The Blues took an early lead against their bottom-of-the-table hosts when Abraham reached Jorginho's pass and guided the ball over Ben Foster.
Watford's goalkeeper denied Abraham, Pulisic and Mason Mount before Chelsea's United States international, who scored a hat-trick against Burnley last weekend, turned home Abraham's cross 10 minutes into the second half.
Gerard Deulofeu scored an 80th-minute penalty, one that was only awarded by VAR (video assistant referee) after a lengthy delay, but Frank Lampard's side held on to climb above Leicester City in the table.
Watford was up against it after just five minutes when Abraham registered his ninth league goal of the season, although it was his first in five games for the Blues.
The England striker latched on to a brilliant first-time throughball from Jorginho, capitalised on hesitation between Craig Dawson and Foster and poked his finish over the Watford goalkeeper.
Abraham had a great chance to add a second when a deflected Mount shot fell kindly to him, yet he was thwarted by Foster, who then did brilliantly to tip Pulisic's header behind.
Deulofeu fired Watford's first chance of note wide and the Hornets were fortunate to only be one goal behind at the break after Foster pushed Mount's driven effort onto the crossbar.
The goalkeeper denied Mount again early in the second half too, though he was unable to prevent Pulisic turning in Abraham's cross for his fourth goal in eight days.
Pulisic nearly had another but Foster flicked out his left foot and the Hornets had hope when they were awarded a penalty 10 minutes from time.
Referee Anthony Taylor was unmoved when Deulofeu went to ground after Jorginho flicked a leg out and though the Spaniard appeared to take another step after the contact before going to ground, the spot-kick was awarded after a VAR review.
There was nearly a dramatic conclusion to proceedings when the brilliant Foster, up in Chelsea's box for a free-kick, headed towards goal but Kepa Arrizabalaga denied him in the final act as the visitors held on.