Both sides were level at the break but Van Persie converted a 62nd-minute penalty before Rooney, who had just signed a bumper new deal at Old Trafford, doubled the visitors’ lead six minutes later.
The result moves David Moyes’ side level on points with the Scot’s former side, Everton, with United moving ahead in the Premier League table on goal difference.
Rooney pledged his long-term future to the Old Trafford club by agreeing a new five-and-a-half-year deal on Friday and he showed his value to the champions with his 12th goal of the season in all competitions 24 hours later as United ended their three-match winless run.
For Tony Pulis’ Palace, the defeat ends their run of three straight top-flight wins at home and keeps them firmly in the relegation picture.
Adnan Januzaj – part of a front four that also included Rooney, Van Persie and Juan Mata – threatened early on for the visitors, dragging a shot wide of Speroni’s left-hand post.
And United enjoyed the better of the opening 15 minutes, keeping the ball well but failing to get in behind Palace until Januzaj jinked past Joel Ward – with the help of an arm – before firing home a strike that was ruled out by referee Michael Oliver.
But despite their array of attacking talent, Moyes’ men struggled to break down a Palace side that came into the game unbeaten at home in the league this calendar year.
Rooney saw his delivery cleared off the line by Damien Delaney before Nemanja Vidic’s follow-up flew over the crossbar.
At the other end, David De Gea was called into action just before the half-hour mark, getting down to his right to save Ward’s deflected header.
Marouane Fellaini, making his first appearance since December, wasted an ideal opportunity to hand Moyes’ men a half-time lead but he curled his effort over the crossbar after a burst into the penalty area.
Jason Puncheon could also have done better with Glenn Murray’s knockdown after the break – with the former Southampton man only managing a tame effort that De Gea easily smothered.
Murray then went close with an acrobatic effort just before the hour but, when Van Persie coolly fired home from the spot after 62 minutes, it took the wind out of Palace’s sails somewhat.
The Dutchman swept home his spot-kick, after Chamakh had clumsily taken out Patrice Evra in the area, before Rooney doubled United’s lead with 22 minutes remaining.
After Evra’s cross sat up nicely for him, Rooney cut across the ball, diverting it past Speroni and into the Argentinian’s top left-hand corner.
The England international’s goal put the gloss on a much-needed United victory and serves Moyes’ men as ideal preparation for their Champions League clash with Olympiakos on Tuesday.