McAnuff's 79th minute strike earned Wimbledon their first points on the road this season after a run of seven successive away defeats.
His first goal of the season left Albion reeling - and sentenced them to a second home defeat in three successive home games after a run a six straight wins at the Hawthorns.
While McAnuff's goal grabbed the headlines, it was the display of Wimbledon goalkeeper Paul Heald that also frustrated Albion.
He pulled off a series of top class saves to lay the foundations for Wimbledon's shock win.
Delighted Wimbledon boss Stuart Murdoch said: "I asked for a response after we lost 6-0 at Nottingham Forest on Saturday and I got it.
"The players showed their character. It was not pretty, but we have been pretty this season and got nothing.
Albion would have been well in control of the match before half-time if it had not been for Heald who pulled off four stunning saves.
He dived low to his right to hold on to a close-range header from Bernt Haas after 11 minutes.
Striker Scott Dobie should then really have broken the deadlock in the 28th minute when he found himself just six yards from goal after a pass from Haas.
But Scottish international Dobie could not find a way past Heald who spread himself on the line to smother the shot.
Heald's heroics threatened to pay a rich dividend when Wimbledon made their one attack of the first-half after Nigel Reo-Coker surged forward from midfield.
He outpaced the Albion defence only to drag his 36th minute shot wide.
Albion heeded the warning and launched an onslaught on the Wimbledon goal only for Heald to stand firm.
He twice denied central defender Sean Gregan who moved forward to fire in two long-range drives.
Heald also pushed a close-range header from centre half Thomas Gaardsoe over the bar.
When Koumas whipped a free kick just wide after 47 minutes it seemed only a matter of time before Albion would strike.
But after Heald pushed away a cross shot from Lee Hughes in the 63rd minute, Albion's failings in front of goal came back to haunt them.
Reo-Coker and Wade Small combined on the right wing before Small whipped in a cross for substitute Pat Agyemang.
His shot was deflected into the path of McAnuff who drilled home his match winner from 16 yards.