Damien Duff scored after just three minutes as United struggled again in defence following their opening defeat at Everton, but Van Persie drew his new employers level with a brilliant finish after 10 minutes, the former Arsenal captain caressing a volley into the far corner.
With Rooney unexpectedly dropped to the bench, Kagawa, a summer recruit from Borussia Dortmund, added a second from close range before the adventurous Rafael made it three with a header to give United a comfortable lead at half time.
Fulham battled on bravely, however, and their ongoing endeavour was rewarded when David De Gea failed to deal with a cross from the left and Nemanja Vidic turned the ball into his own net.
But while United clung on to record their first win of the season, they lost Rooney to a nasty gash on the leg in a nervy period of injury time. His thigh was opened up by the boot of Hugo Rodallega and Rooney was taken off on a stretcher, with initial estimates suggesting the striker could be absent for as much as four weeks.
Rooney was initially punished for a rather listless performance in the defeat to Everton when being dropped by Ferguson for the first time since Boxing Day 2011. Indeed, the striker started only two games on the bench for United last season, the arrival of Van Persie apparently immediately threatening his untouchable status at Old Trafford.
Rooney found himself in good company though: the United bench had 1,776 Premier League appearances between them, 110 of which belonged to Jonny Evans. The defender was returning from injury to help ease United's injury problems, but not quite soon enough as Michael Carrick, who was bullied by Marouane Fellaini in Everton's shock 1-0 win on Monday night, found himself exposed once again.
After only two minutes the makeshift centre-back was caught out by Mladen Petric - scorer of two goals on debut last weekend - as he conceded a clumsy free kick right on the edge of the box. United's lack of defensive nous showed again when Bryan Ruiz picked out Duff with a disguised, low pass to the penalty spot and the winger, who this week retired from international football, swept the ball home first time.
United had to respond with a newly calibrated attack that was geared around Kagawa playing off Van Persie. With Rooney benched, they needed the new arrivals to settle in quickly and Van Persie played his part, opening his United account with his first shot for the club after 10 minutes. It was a sumptuous finish from the Dutchman as he met a bouncing cross from Patrice Evra near the penalty spot with a gorgeous volley to turn the ball inside the far corner.
Despite the accomplished nature of the finish, United were lacking a bit of fluency initially, though Antonio Valencia was a constant threat having been restored to his favoured position at right midfield and linked up well with right-back Rafael, who was keen to push forward at every given opportunity and attack the Fulham left.
Kagawa, stationed centrally, began to grow in influence and after having one shot and playing in Ashley Young for another, the Japan international joined Van Persie in scoring his first United goal after 35 minutes. The Dutchman had a corner cleared as far as Tom Cleverley and when the midfielder's low effort was spilled by Schwarzer, Kagawa was on hand and onside to tap the ball home.
Fulham were restricted to a sniff of a half chance when De Gea indulged in some inadvisable skills in his own box, dragging the ball through his legs and away from the charging Petric following a Rafael back-pass.
For the first time in this embryonic season there seemed a real swagger about United and few players embodied this as graphically as Rafael, who decided to abandon any pretence at defence and perform as an auxiliary attacker. Indeed, just minutes after having a goal ruled out for offside when he tapped in following a shot by Kagawa against the post and a follow-up effort from Young, he extended United's lead with a firm header at the back post from a chipped Young cross.
Fulham should have pulled one back just minutes before the break when Alexander Kacaniklic had a low shot saved by De Gea, and Petric then saw his follow-up effort deflected onto the bar by the desperate Spanish keeper. But United retained a two-goal lead conjured up by their two big summer signings.
Half-time presaged a dip in United's performance though. They continued to dominate possession but Fulham became harder to break down, Sascha Riether and Hangeland in particular proving a formidable barrier to United's aspirations. When the visiting defence was opened up by a fine cross from Van Persie, Rafael's misdirected header took the ball away from the more advantageously placed Young at the back post
After 64 minutes, United's veneer of superiority was punctured as Fulham narrowed their lead to a solitary goal. It was a regulation cross from the left from Fulham but De Gea flapped at the high ball and the muscular presence of Petric was enough to put Vidic off balance and divert the ball into his own net with his heel.
De Gea looked accusingly at his captain following the incident but he should have dealt with the situation far better. Though undoubtedly a very talented goalkeeper who makes plenty of fine saves, there remains a lingering uncertainty about the Spaniard that has proved hard to shake.
He showed the more admirable side of his nature soon after when tipping a firm effort from Moussa Dembele over the bar and then made an even more impressive save when denying the same player from close range after United comprehensively failed to deal with the subsequent corner.
The home side were keen to kill the game off as a contest and Rafael almost did so when seeing a superb effort turned over the bar by Schwarzer. However, Fulham always stayed in touch and almost drew level deep into injury time.
United found themselves down to 10 men when, with having made all three changes, Rooney sustained the lengthy gash on his thigh after Rodallega accidentally landed on him after a challenge.
Fulham had one last chance and the impressive Dembele swung a cross into the box to pick out the head of Ruiz. However, De Gea was equal to the forward's header as he kept the ball out to ensure United picked up their first three points of the season.