Ashley Young's cross led to the opener but the visitors grew in stature in the second half and ensured a nail-biting afternoon for the Old Trafford faithful until substitute van Persie sealed another important victory.
As he'd promised, Sir Alex Ferguson tinkered with his line-up, making five changes, with Shinji Kagawa returning to the side behind Danny Welbeck. Thankfully, the rain had that prompted a pre-match pitch inspection abated but the soggy surface necessitated the Reds getting the ball forward quicker than usual.
After a shaky opening - during which Michael Carrick needed to slide-tackle Shane Long after the Irishman had turned Nemanja Vidic and Chris Smalling was fortunate not to concede a penalty after a rash challenge on the lively Baggies attacker - United took control of proceedings.
Young worked a one-two with Kagawa and, when the ball fell into his path, the winger drilled in a cross that deflected off Gareth McAuley and past former Reds keeper Ben Foster in front of the jubilant Stretford End.
The Barclays Premier League leaders looked to extend the advantage with Welbeck unable to reach a long ball by Tom Cleverley and then working Foster from a Kagawa pass. Some of the football was spellbinding at times and, after one stunning move, Carrick had a shot charged down.