Rooney scored his 249th United goal in the seventh minute at Old Trafford, prompting applause from the watching Charlton, before an Anthony Martial goal and a Marcus Rashford brace completed victory.
Third-tier Millwall pulled off the biggest shock of the day by beating top-flight Bournemouth 3-0 at The Den, while Stoke City and West Bromwich Albion both fell to second-tier Championship opposition.
"It's a proud moment, to do it at a massive club like Manchester United," Rooney, 31, told BT Sport after matching Charlton's mark on his 543rd United appearance.
"I'm hugely honoured to play for this club. To be up there with Sir Bobby in terms of goals is a huge honour. Hopefully I'll be up there on my own soon."
It was to prove a sorry return to Old Trafford for Reading manager Jaap Stam, the former United centre-back.
Making his first appearance since December 17, Rooney broke the deadlock when he used his right knee to steer Juan Mata's dinked cross past Reading goalkeeper Ali Al Habsi.
Rooney created United's second goal in the 15th minute by playing a one-two with Martial, who stroked a cool effort into the bottom-right corner.
Rashford made the game safe late on, running onto Michael Carrick's pass to score and then pouncing to lash home after Al Habsi embarrassed himself by completely miscuing an attempted clearance.
Rooney has now surpassed Charlton, 79, as the leading scorer for both United and England.
United, meanwhile, have won eight successive games in all competitions for the first time since 2009.
"I am happy because it was a professional job from minute one," said United manager Jose Mourinho, whose side host Hull City in the first leg of their League Cup semi-final on Tuesday.
"Bobby Charlton is Bobby Charlton, but to have Wayne as the top scorer in such a big club is great for him."
- Leicester comeback -
United join cross-town rivals Manchester City in round four after Pep Guardiola's side romped to a 5-0 victory at West Ham United on Friday.
Steve Morison, Shaun Cummings and Shane Ferguson scored as London side Millwall, beaten FA Cup finalists in 2004, won 3-0 at home to top-tier Bournemouth, whose manager Eddie Howe made 11 changes.
West Brom, riding high in eighth place in the Premier League, lost 2-1 at home to Championship promotion-chasers Derby County, with Tom Ince netting a 54th-minute winner.
Portuguese winger Helder Costa, on loan from Benfica, and Matt Doherty scored in each half as Wolverhampton Wanderers won 2-0 at Stoke.
Southampton face a replay against second-tier opposition after Steven Naismith scored in the 92nd minute to earn Norwich City a 2-2 draw.
Ahmed Musa scored twice in five minutes as misfiring Premier League champions Leicester City -- currently 15th in the top flight -- came from behind to win 2-1 at Everton, who had gone ahead through Romelu Lukaku.
Marco Silva won the battle of the new Premier League managers, late goals by Abel Hernandez and Josh Tymon giving Hull City a 2-0 victory over Paul Clement's Swansea City.
Crystal Palace manager Sam Allardyce was frustrated on his return to former club Bolton Wanderers as the League One team held the south London side to a 0-0 draw.
Watford beat Burton Albion 2-0, while the all-Premier League tie between Sunderland and Burnley finished goalless.
There was no fairytale for seventh-tier Stourbridge, the lowest-ranked team in the competition, who lost 2-1 at Wycombe Wanderers of League Two.
Arsenal beat Preston 2-1
Later in the day, Arsenal were forced to stage a second half fightback to secure a 2-1 win at Preston on Saturday that extended Arsene Wenger's record of never having lost in the FA Cup third round.
Wenger's side suffered an early shock at Deepdale as Callum Robinson put the Championship side on course for their first victory over the Gunners since 1960.
Arsenal were fortunate to be only one goal down at the interval, but they produced a much-improved second half showing.
Aaron Ramsey got the equaliser less than a minute into the second half and Olivier Giroud completed the comeback in the final minute.
Preston, buoyed by a fervent home crowd, took the lead after only seven minutes to ensure a pulsating cup tie.
Aiden McGeady showed neat footwork in midfield, dragging the ball back and spinning to create space to beat his marker before slotting a pass inside Nacho Monreal for Jordan Hugill.
McGeady was ushered off the ball by Monreal but it broke kindly for Robinson, who took advantage of some leisurely defending to pounce on the ricochet and steer past the on-rushing David Ospina.
Despite fielding a strong side, Wenger must have been disappointed to see his side struggle to cope with the early pace being set by Simon Grayson's team.
Indeed, the only surprise of the opening half was that Arsenal were not trailing by more than a solitary goal.
On 22 minutes McGeady again found space behind the defence to play in Hugill whose cross was met by a strong Robinson header, needing a block from Shkodran Mustafi to deny a goal.
Seconds later, it was Robinson's turn to be the creator, crossing low from the right for the unmarked Hugill who somehow failed to make contact in front of an open goal for what would have been a certain second.
By the time the excellent McGeady crossed from the left and full-back Marnick Vermijl arrived unmarked to volley wide, the Arsenal supporters behind that goal were beginning to voice their disapproval.
- Whole-hearted -
Giroud had Arsenal's first meaningful chance on the half-hour, glancing well wide from Ainsley Maitland-Niles' cross, but at least there were signs of Arsenal life.
In quick succession, Lucas Perez and Ramsey should have done better as they tried to walk the ball into the net, only to be blocked by whole-hearted Preston defending.
Giroud chipped across the face of goal when he might have shot, while Perez launched a powerful header that was well held by Chris Maxwell.
Still, there were warning signs for Wenger as a Mustafi error allowed Gallagher a chip which cleared the goalkeeper and was bound for the net until Monreal made an appearance to clear.
Wenger must have delivered a half-time team talk of some impact because, within 55 seconds of the restart his side was level.
Alex Oxlade-Chamberlain launched the attack down the left and, after picking out Alex Iwobi in the Preston area, the ball was worked to Ramsey who struck an unstoppable shot past Maxwell from 18 yards.
Giroud was also handed a chance, from Granit Xhaka's well-judged 65th minute free-kick, but only managed to head directly at Maxwell under pressure from more dogged Preston defending.
And Mustafi was also denied by Maxwell, with a far more impressive reflex save, from a near-post snapshot following an Arsenal corner.
Not for the first time this season, it was the often maligned Giroud who came to Arsenal's rescue with another late goal, this time in the 89th minute when he forced home from six yards after Perez managed to clip the ball back from the byline.
To complete what had become a satisfactory afternoon for Wenger, the Arsenal manager was able to bring on England forward Danny Welbeck for the final seven minutes for his first appearance since damaging knee ligaments eight months ago.