And after going close twice, Anthony Martial ended his 491-minute goal drought by collecting Paul Pogba's pass and beating Ramsdale at the second attempt.
Griezmann and Morata both scored again in the second half, either side of Samuel Lino's stunning strike just minutes after he came off the bench. Saul Niguez added a sixth late on.
Chelsea had the better of the second half but could not carve out the chance to score a potentially vital second. Manager Tuchel will still be very satisfied with both the performance and result.
A dinked pass intended for De Bruyne from the impressive Oscar Bobb then deflected off Jackson and looped into the net early in the second half before Foden smashed in his second.
The Spaniard then set up Lemina for the winner in the 97th minute, seeing Spurs slip to their second straight defeat and miss the chance to replace Manchester City at the top of the Premier League.
And following two more spurned chances from Oscar, Costa arrived on the scene in the 81st minute to score his second and haul Chelsea level, firing home an overhead kick after Swansea had failed to deal with the danger in the box.
In a tight and cagey second leg, Gabriel Martinelli had Arsenal's best chance of the first half but shot straight at Manuel Neuer after Jamal Musiala had forced David Raya into a full-stretch save at the other end.
The visitors looked threatening on the break all game, with Jackson earlier spurning an even better chance from Malo Gusto's cross when a heavy touch allowed Ederson to smother his shot, and City's keeper also denied Sterling a second from point-blank