Manchester City maintained a one-point Premier League lead over Liverpool with four games to go after seeing off Leeds 4-0 on Saturday as Norwich were relegated after a 2-0 loss at Aston Villa.
Earlier, Liverpool had moved to the top of the table with a 1-0 win at Newcastle thanks to Naby Keita's first-half goal.
Both the title contenders each made five changes with tricky away trips sandwiched between both legs of their Champions League semi-finals.
City boss Pep Guardiola could afford the luxury of not even calling on Kevin De Bruyne as the champions kept the fate of the title race in their own hands.
Rodri headed home Phil Foden's free-kick on 13 minutes to settle any City nerves.
Nathan Ake then turned home Ruben Dias' knockdown as the visitors struck again from a set piece early in the second-half.
Gabriel Jesus kept his fine scoring form going with a thumping finish 12 minutes from time before Fernandinho rounded off the scoring in stoppage time.
"It's not about the pressure – it's simple," said Guardiola.
"We have to win all our games to be champions. If we don't, Liverpool will be champions."
Liverpool's top scorer Mohamed Salah, Thiago Alcantara, Fabinho and Trent Alexander-Arnold were among those rested by Jurgen Klopp from the start.
But they still had far too much for one of the form sides in the league.
The decisive moment came in the 19th minute when, with the home side appealing in vain for a free-kick, Jordan Henderson and Diogo Jota combined to feed Keita, who stepped inside before dispatching the ball past the helpless Martin Dubravka.
The Reds could have made life easier for themselves but Sadio Mane and Jota squandered chances and they survived a late scare when Alisson Becker saved from Bruno Guimaraes.
Klopp said he was delighted with his players' ability to produce a winning performance after so many changes as they chase sporting immortality in an unprecedented quadruple.
"I thought it was a top-class game in extremely difficult circumstances," said Klopp, who had voiced his anger at being handed a lunchtime kick-off just three days after facing Villarreal.
"That's why I'm really happy. You want to win football games and in the best case you deserve to win. We did today 100 percent against a team in great form in a great atmosphere."
Norwich are heading back to the Championship after just one season in the top flight after goals from Ollie Watkins and Danny Ings sealed their fate on manager Dean Smith's return to Villa Park.
"I'm disappointed with our result," said Canaries captain Grant Hanley. "It is probably fair, we haven't been good enough this season."
Burnley's turnaround to beat Watford 2-1 at Vicarage Road sealed the Canaries fate.
The Clarets looked to be heading towards the trap door when they sacked long-serving boss Sean Dyche two weeks ago but have taken 10 points from a possible 12 since.
James Tarkowski's early own goal gave the visitors an uphill task but late goals from Jack Cork and Josh Brownhill turned the match on his head.
"Credit to the boys, the staff, everybody – we have rallied each other and we want to be in this division," said Brownhill. "We are hard-working and we deserve to be in this division."
The result means Watford are all but doomed while 18th-placed Everton are in desperate trouble – five points from safety and facing playing outside the English top-flight next season for the first time since 1954.
Brighton hammered Wolves 3-0 and Crystal Palace beat Southampton 2-1.