He netted his hattrick in the 64th minute, tapping home at the far post after some superb build-up play from the home side. The CR7 show continued in the 76th minute as he scored his fourth.
Julian Speroni kept Palace in the game with a series of fine saves before Darmian doubled the lead by lashing homefrom 20 yards. Arsenal, who have a game in hand, entertain West Brom on Thursday.
PSG were farfrom their fluent best on Sunday lunchtime but the French champions did enough to secure a nervy 1-0 win away at Ligue 1 surprise package Le Havre.
Mauricio Pochettino's side were far more clinical after the break to take the game awayfrom Preston as the visitors tired. Chelsea can now turn their attention to the first leg of their Carabao Cup semi-final at Middlesbrough on Tuesday.
Ramsey had better luck half an hour in, though, sweeping a left-foot shot awayfrom Martin Dubravka and in off the right-hand post with the first registered shot of the match.