Luis Suarez slotted in the Reds' first in the 14th minute from Jose Enrique's pass and Daniel Sturridge doubled the advantage with a fierce strike three minutes later.
With seven minutes remaining before the interval, Steven Gerrard then scored a penalty, given for a Dean Moxey foul on Raheem Sterling.
Substitute Dwight Gayle headed in a consolation goal for second-bottom Palace in the 77th minute.
City went into the contest off the back of last weekend's 3-2 top-flight defeat at Aston Villa and Wednesday's 3-1 Champions League home loss to Bayern Munich, and they were a goal down in the 16th minute when Romelu Lukaku scored.
A minute later Alvaro Negredo equalised and Sergio Aguero put the hosts 2-1 up just prior to half-time, before the unlucky Tim Howard diverted in City's third in the 69th minute.
The Everton goalkeeper tipped Aguero's penalty - awarded for Seamus Coleman's challenge on Pablo Zabaleta - onto a post but the ball hit Howard's head on the rebound and went in.
The 18-year-old Belgian winger, who had earlier been booked for diving, dragged a misfiring United back into the game with a calmly-taken 55th-minute equaliser and then smashed a dipping volley past keeper Keiren Westwood to win the game just six minutes later.
On an afternoon when United were far from their best, they were all but taken apart by the bottom-of-the-table Black Cats before the break with Craig Gardner's fifth-minute opener setting the stage for a concerted assault by the home side.
But David Moyes' men, who were facing a third successive Premier League defeat for the first time since December 2001, responded in the second half and eventually earned their reward.
The three points were sealed by Darren Bent's left-footed strike with seven minutes of normal time to go.
Newcastle, who had suffered losses in their previous two league outings, also bounced back as they won 2-1 at Cardiff, with Loic Remy netting both Magpies goals to take his tally to five in his last three matches.
Remy drilled in his first on the half-hour mark having burst forward and then converted on the follow-up in the 38th minute when David Marshall parried Papiss Cisse's shot.
The home side pulled a goal back 13 minutes after the break through Peter Odemwingie, but Newcastle were able to hold on.