City had to fight back from 2-0 down in the second half, but three goals in the space of five minutes — including two from German substitute Ilkay Gundogan set off the party, with fans pouring onto the field at the Etihad Stadium at the final whistle.
Liverpool had also found it hard to get their business done at Anfield against Wolverhampton Wanderers but late goals from Mohamed Salah and Andy Robertson earned them a 3-1 win.
Tottenham Hotspur comfortably secured the fourth Champions League spot with a 5-0 win at relegated Norwich City while North London rivals Arsenal had to settle for fifth spot despite crushing Everton 5-1 at the Emirates Stadium.
Burnley were relegated after they lost 2-1 at home to Newcastle United while Leeds United secured survival with a 2-1 victory at Brentford.
Norwich and Watford had already been relegated before the final day. (Reuters)