The rain-soaked pitch, which was subject to two inspections prior to kick-off, put paid to any hope of free-flowing football but McAuley's blunder and substitute Van Persie's sublime finish were enough to separate the sides.
After having a good shout for a penalty turned down in the opening few minutes by referee Jonathan Moss, when Shane Long went down under a challenge from Chris Smalling, Baggies defender McAuley inadvertently gave United an early lead.
Ashley Young's attempted one-two with Shinji Kagawa was diverted back into the winger's path by the outstretched leg of Graham Dorrans, allowing Young time and space to drill a low cross into the danger zone, which cannoned off the despairing McAuley to put the league leaders in front.
Ben Foster kept his side in the match twice in quick succession just after the half-hour mark, firstly brilliantly tipping Young's powerful effort over the top, and then scrambling a Michael Carrick effort away from the resulting corner.
The early goal and a much-changed line-up seemed to affect the Baggies' momentum, as they never really got going in the first period.
However, they made a much brighter start to the second half, with Chris Brunt coming close from distance, and McAuley had a great chance to make amends for the earlier own goal, but could not direct his header on target from a Brunt free-kick.
As the game wore on, the pitch deteriorated even more, stifling much of the play. Kagawa, making his long-awaited return to the side, was the only one whose quality was not hampered by the damp conditions.
He was at the heart of everything for the Red Devils before being replaced by Robin Van Persie.
Another substitution made a huge difference to the game. Romelu Lukaku added an extra dimension to West Brom's attack, with his physical presence causing the United back-line problems. The defending was desperate at times, with some last-ditch blocks keeping the score at 1-0.
However, despite all their pressure the Baggies rarely troubled David De Gea in the United goal, and in fact the league leaders came closest to adding to the scoreline through Van Persie and a Danny Welbeck effort, before the Dutch hitman once again got his name on the scoresheet in added-time.
His stunning left-footed strike from the edge of the area beat Foster all ends up to seal the three points for the Red Devils.