While defending champions and near-neighbours City achieved a come-from-behind win against Southampton on Sunday the Red Devils were second best for most of the contest at Goodison Park.
The thorn in their side was midfielder Marouane Fellaini, who was a constant threat and deservedly capped an impressive performance with the only goal early in the second half.
Fellaini had already headed over Leighton Baines' corner before he collected a throw-in and turned into the penalty area, holding off makeshift centre-half Michael Carrick, to hit the post from the narrowest of angles.
United goalkeeper David De Gea had to be on excellent form and denied Nikica Jelavic, tipped over a long-range Steven Pienaar header and flung himself to his left to save the South Africa midfielder's low shot before preventing Baines' deflected free-kick sneaking into the top corner.
But it was not all one-way traffic as United had chances of their own, though not as good or as numerous as their hosts. Danny Welbeck, the player with most to lose from Van Persie's arrival, had one shot blocked by Phil Jagielka and hooked another wide under pressure from the centre-back.
Fellaini's ravaging of the United defence continued after the interval as his header across goal was smashed against the crossbar by Leon Osman with De Gea, for once, beaten.
And the big Belgian was finally rewarded for his outstanding display with a goal in the 57th minute when former Red Devil Darron Gibson swung over a corner and he left Carrick sprawling on the turf to power home a header.
United's summer signing Shinji Kagawa had plenty of good touches around the penalty area, the best of which in the second half teed up Welbeck who was denied by Jagielka's perfectly-judged tackle.
The Everton defender's timing was even better when he cleared Tom Cleverley's shot off the line before, in the 68th minute, Van Persie was finally sent on.