Abraham ended Palace's dogged defensive resistence with a simple finish from Willian's deft through ball seven minutes into the second half before Pulisic headed home a loose ball 11 minutes from time to seal all three points.
Chelsea edge a point clear of Manchester City and to within five of leaders Liverpool, who host City in a highly-anticipated clash on Sunday.
A 4-4 draw with Ajax on Tuesday took the tally of goals Chelsea's prior 18 games this season to 70 at an average of nearly four per game.
However, the Blues free flowing football under Frank Lampard was met with a brick wall of a Palace defence for the first 45 minutes.
Pulisic had scored four goals in his last two Premier League games as the American begins to deliver on his £58 million ($74 million) price tag and only a fine save from Vicente Guaita denied the USA captain another strike early on.
Thereafter, Chelsea were largely restricted to efforts from free-kicks on the edge of the box before the break as Willian curled inches wide.
Gary Cahill was making his return to Stamford Bridge after his seven years as a Chelsea player came to an end in the summer.
And Cahill reminded his former employers of why he won eight trophies for the club with a brilliant block to repel Willian's goalbound effort in first half stoppage time.
The Palace resistence was finally broken by one moment of genius from Willian as Chelsea's captain for the day flicked the ball perfectly into Abraham's path to slot home his 11th goal of the season.
Pulisic tested Guaita again with a rising drive shortly after that the Spaniard did brilliantly to tip over.
Palace's problem this season has been scoring goals, but they missed a huge chance to snatch a point 20 minutes from time when James Tomkins headed wide from Luka Milivojevic's free-kick.
Instead at the other end, Pulisic finally got the goal his performance deserved by following up after Michy Batshuayi's shot had been blocked to nod past a helpless Guaita.