The only sour note on a straightforward assignment for Chelsea came in the second half, when Ruben Loftus-Cheek limped out of the action with what appeared to be an ankle injury.
Victory ensured Chelsea moved into third in the Premier League, two points clear of Tottenham and three ahead of fifth-placed Arsenal in the race for a top-four finish.
Ross Barkley's 96th-minute equaliser helped Chelsea maintain an unbeaten record under Maurizio Sarri.
Ross Barkley set up Eden Hazard's 30th-minute opening goal before adding the second as Chelsea beat Southampton on Sunday.
The 24-year-old England midfielder has signed a five-and-a-half-year contract and will wear the number eight shirt.
Ross Barkley is out of contract at Goodison Park at the end of this season, after rejecting a new deal with Everton.
Aaron Lennon was on target after 23 minutes while Ross Barkley tucked away two late penalties although Everton should have won more comfortably against Newcastle United, being denied by the woodwork three times in six second-half minutes.
Everton sealed their place in the FA Cup fifth round thanks to goals from Arouna Kone, Aaron Lennon and Ross Barkley.
Ross Barkley, Alex Oxlade-Chamberlain and an own goal by Lithuania goalkeeper Giedrius Arlauskis helped England score a 3-0 win and complete 10 wins out of 10 in their Euro 2016 qualifying campaign.
With a 3-1 win, Everton moved above Merseyside rivals Liverpool into 10th place, while QPR remain in the relegation zone after losing their ninth successive away game in all competitions.
Roberto Martinez has warned Ross Barkley's potential suitors that even a bid equivalent to the astronomical figure paid by Real Madrid for Gareth Bale will not prompt Everton to let the youngster leave Goodison Park.
Ross Barkley stepped up in the 84th to beat Gerhard Tremmel with a free kick that came off the underside of the bar on its way in.
The sides' first meeting since Germany's bruising 4-1 win in the last 16 at the 2010 World Cup in South Africa was decided by a 39th-minute header from Arsenal defender Per Mertesacker.
Spurs had just taken a 31st-minute lead through Andros Townsend at Villa Park when a canister trailing black smoke was lobbed towards the playing surface, apparently from the away supporters' section.
Roberto Soldado added a second after half-time as Spurs moved up to fifth place in the table, three points below leaders Arsenal, and began to atone for their shock 3-0 home defeat by West Ham United.
Prior to this season Townsend, who had scored three goals in 18 Spurs appearances, had had loan spells at nine different clubs.
According to the reports, England manager Roy Hodgson encouraged his players to pass the ball to in-form Tottenham Hotspur winger Townsend by telling a joke about a monkey being sent into space by US space agency NASA.