This is David Moyes' third successive short-lived stint at a club, following previous disappointments at Manchester United, where he lasted just 10 months, and Real Sociedad.
Arsenal failed to qualify for the Champions League for the first time in 20 years after Manchester City and Liverpool wrapped up the remaining top four places on the final day of the Premier League season.
Sergio Romero displayed some eye-catching pre-Europa League final form by saving a penalty as Manchester United drew 0-0 at Southampton in their penultimate Premier League game.
Alexis Sanchez sank Sunderland to keep Arsenal in the hunt for Champions League qualification, while Manchester City all but secured a top-four place by beating West Bromwich Albion.
Skipper John Terry, who will leave Chelsea at the end of the season, crowned his 716th game for the club with the opening goal, but was then at fault as Etienne Capoue equalised.
Liverpool are on the brink of qualifying for the Champions League after Philippe Coutinho inspired a 4-0 rout of West Ham, while Tottenham said a fitting farewell to White Hart Lane with a 2-1 win against Manchester United on Sunday.
Giroud scored the opening goal on 42 minutes as Arsenal scythed their way through the Stoke defence impressively.
Machester City went past Leicester City 2-1 as Swansea beat Sunderland 2-0 to boost their survival bid while Southampton ended their barren run against Middlesbrough on Saturday.
It is Chelsea's second league title win in three seasons, their sixth in total, and comes at the end of former Juventus and Italy coach Antonio Conte's first season as manager.
Indian owners Venky's, who purchased Blackburn Rovers when they were still in the Premier League in 2010, have presided over a string of disastrous decisions, from the moment they announced their arrival by sacking of manager Sam Allardyce.
The win over Middlesbrough left Chelsea seven points clear of Tottenham Hotspur at the summit and means they will be crowned Premier League champions for the second time in three seasons if they win at West Bromwich Albion on Friday.
Arsenal are back in the hunt to qualify for the Champions League after Arsene Wenger finally got the better of Jose Mourinho in a 2-0 win over Manchester United, while Liverpool were held to a 0-0 draw by Southampton on Sunday.
Manchester City leapfrogged Liverpool on goal difference into third place and also moved four points clear of fifth-placed Manchester United as the battle to finish in the top four heats up.
Having seen top-four rivals Manchester City, Manchester United and Arsenal all drop points on Sunday, Liverpool capitalised thanks to Can's jaw-dropping effort in first-half injury time.
Dele Alli and Harry Kane kept Tottenham Hotspur in the Premier League title hunt as their side torpedoed Arsenal 2-0 while Chelsea maintained their advantage by winning 3-0 at Everton earlier in the day.
Booked for tripping Aguero in the 84th minute, Fellaini fouled him again seconds later and then blatantly butted the Argentinian, leaving referee Martin Atkinson no option but to send the Belgian off.
Leaders Chelsea, who beat Tottenham 4-2 in the FA Cup semi-finals on Saturday, went seven points clear by beating Southampton on Tuesday, but Spurs' win trimmed the gap back to four points.
Costa set up Eden Hazard's early opener, which was cancelled out by Oriol Romeu, before Gary Cahill restored the hosts' lead at an unseasonably cold Stamford Bridge.