The left-back smashed one in from the edge of the area before converting a penalty - his 10th success from the spot to maintain his 100% Premier League record.
Substitute Shane Long pulled a goal back midway through the second half but the Baggies, who had won just once in their previous 13 league visits, saw their recent slump extended to just one point from a possible 15.
The opening goal virtually came out of nowhere - which was precisely where West Brom's defence were - when Baines picked up Leon Osman's pass. With no tackle forthcoming the left-back advanced to the edge of the area before smashing a shot past Ben Foster.
West Brom had George Thorne carried off on a stretcher just before half-time after a collision with Anichebe and it proved to be a double blow. Substitute Claudio Jacob's first significant involvement was to concede the penalty just four minutes after coming on.
Steven Pienaar's backheel released Anichebe on the left of the area and the Argentina midfielder, who has missed most of the last eight weeks with injuries, brought him down. With a man like Baines on spot-kick duty it seems you are guaranteed a goal - and the 28-year-old did not disappoint by smashing home his second.
The introduction of Long for James Morrison in the 62nd minute paid immediate dividends as within three minutes he had pulled one back. Romelu Lukaku's diagonal run saw him break clear of the defence and round goalkeeper Tim Howard but his cross-shot was bundled up into the air by John Heitinga and Long reacted quickest to head in.
Finally West Brom found another gear and when Graham Dorrans rattled the crossbar with a long-range free-kick they began to sense their chance.
Pienaar's curling free-kick was tipped over from darron Gibson and he South African's next contribution should have brought the hosts' third goal but when his deflected low shot rebounded off the post Osman blazed over the open goal from six yards.