A controversial goal from Anthony le Tallec handed Sunderland only their second win of the season.
Greening takes on the Sunderland defence
A dreadful game burst into life after 72 minutes when Liam Lawrence seemed to push Darren Carter to the floor.
But referee Phil Dowd played on and Le Tallec's shot was heading wide before taking a deflection off Steve Watson and wrong-footing Tomasz Kuszczak.
West Brom spent much of the last 15 minutes in the Sunderland half but rarely looked like pinching a point.
Sunderland failed to muster a shot on target before the goal but were rewarded for a battling performance.
West Brom were well below par, although they did at least manage to bring a save out of Sunderland keeper Kelvin Davis.
That came from a shot by Ronnie Wallwork that was going wide until an inadvertent deflection from Junichi Inamoto brought a save from Davis.
Watson should have tested Davis when he took a free-kick on his chest and had all the time in the world to shoot, but for some reason he squared it for Nathan Ellington to fire over.
Sunderland carved out their first real opening five minutes after the break when Lawrence picked out Arca with a perfect cross and the left winger should have done better from close range than fire wide.
It sparked a much improved effort from Mick McCarthy's strugglers and as the Baggies poured forward they began to look dangerous on the break.
But Bryan Robson will feel aggrieved at the build-up to the goal, with Lawrence's shove coming right in front of the Baggies boss.
McCarthy will have no complaints, having seen his side denied three points with a last minute goal in the game between the sides on Wearside earlier this season.
West Brom boss Bryan Robson:
"I thought the first half wasn't too bad. But the second half performance was very disappointing. I don't know what happened with the boys.
"There were quite a few of them who were well below par, they didn't take any responsibility on the ball.
"I knew it would be difficult, but we made it difficult for ourselves."
Sunderland boss Mick McCarthy:
"I think today we've earned a very good three points - we said in the dressing room the game was there to be won.
"We had to work hard and dig in but it's a justified three points - we never stopped fighting.
"The players are remarkable. Some teams would have thrown the towel in, these players have kept me in a job."
West Brom: Kuszczak, Albrechtsen, Curtis Davies, Watson, Robinson, Greening, Inamoto (Horsfield 69), Wallwork (Chaplow 86), Carter, Campbell, Ellington (Nicholson 79).
Subs Not Used: Kirkland, Clement.
Sunderland: Davis, Hoyte, Breen, Caldwell, Danny Collins, Lawrence (Nosworthy 90), Whitehead, Miller, Arca, Stead (Gray 86), Le Tallec (Bassila 76).
Subs Not Used: Alnwick, Murphy.
Booked: Caldwell, Breen, Miller.
Goals: Le Tallec 72.
Ref: P Dowd (Staffordshire).