The Red Devils recovered from a sluggish start to threaten the Fulham goal through captain Bruno Fernandes, who was twice denied by the smart reactions of Bernd Leno in the first half at Old Trafford.
Mason Mount, who is looking to put an injury-hit season behind him, was then unable to make the most of his golden opportunity in the opening stages of the second period, with his close-range effort producing another fine stop from the defiant Fulham goalkeeper.
Man United boss Ten Hag attempted to make an impact on proceedings by making numerous substitutions throughout the second half, including debut appearances for Zirkzee and Matthijs de Ligt.
Zirkzee's introduction ultimately proved to be a stroke of genius as the former Bologna striker produced the all-important breakthrough in the 87th minute.
The Dutchman timed his run into the box to meet Alejandro Garnacho's inviting delivery, steering a composed first-time finish beyond the reach of Leno.
After grabbing an assist, Garnacho had a glorious chance to get his own name on the scoresheet, but the Argentine conspired to drag the ball harmlessly wide of the Fulham goal.
Fortunately for Man United, Garnacho's late miss did not prove costly as the Red Devils Ten Hag's third season in charge with a narrow victory.
MANCHESTER UNITED VS. FULHAM HIGHLIGHTS
Mount goes close vs. Fulham (53rd min, Man United 0-0 Fulham)
Joshua Zirkzee goal vs. Fulham (87th min, Man United 1-0 Fulham)
Garnacho fluffs his lines vs. Fulham (94th min, Man United 1-0 Fulham)
MAN OF THE MATCH - JOSHUA ZIRKZEE
Bernd Leno deserved an honourable mention after making several impressive saves during the course of the 90 minutes.
However, Leno's performance will not get the headlines as Zirkzee produced the game-defining moment to get Man United's season off to a winning start.
The Netherlands international may have only been introduced on the hour-mark, but he deserved the man of the match award for making an instant impact under the enormous pressure of a Man United debut.
Zirkzee showcased his goalscoring instincts to produce a fine first-time finish to get the privilege of scoring the first goal of the 2024-25 Premier League season.
MANCHESTER UNITED VS. FULHAM MATCH STATS
Possession: Man United 55%-45% Fulham
Shots: Man United 14-10 Fulham
Shots on target: Man United 5-2 Fulham
Corners: Man United 7-8 Fulham
Fouls: Man United 12-10 Fulham
BEST STATS
WHAT NEXT?
After snatching a late win, Man United will target a second consecutive victory when they travel to the Amex Stadium for a meeting with Brighton & Hove Albion next weekend.
As for the visitors, they would have been disappointed to concede late on, but they can still take some encouragement from their battling performance as they now turn their focus to their home meeting with newly-promoted Leicester City.