The hosts burst out of the blocks at The Hawthorns and took a deserved lead thanks to Craig Dawson's powerful 10th-minute header.
West Ham's character was on display for all to see, though, as captain Kevin Nolan - drafted in to replace Mark Noble who has a calf problem - nodded in the equaliser after 35 minutes.
Buoyed by his side’s impressive, albeit fruitless, display against Arsenal on Saturday, West Brom boss Alan Irvine included Saido Berahino and Anichebe from the start in an attacking line-up, while Chris Baird replaced the injured Sebastien Pocognoli at left-back.
Baird's free-kick fell invitingly for Nigeria international Anichebe in the seventh minute, but his effort was deflected over, before a succession of set-pieces and West Brom pressure ultimately led to Dawson's opener.
Graham Dorrans' well-placed free-kick was met powerfully by the head of the defender, with West Brom goalkeeper Adrian, who came off his line, well beaten.
Nolan's bicycle kick was well saved by Ben Foster before West Ham levelled 10 minutes from the interval, as the rebound from Andy Carroll's fierce overhead effort - which was also kept out by the West Brom goalkeeper - was headed in from close range by the skipper.
Sam Allardyce's men turned the screw and Cheikhou Kouyate had a curling effort turned behind by Foster, while Chris Brunt hobbled off for the hosts, replaced by Youssouf Mulumbu.
And West Ham capped off their first-half turnaround when, in the third minute of stoppage time, Tomkins climbed above Dawson at the back post to nod Stewart Downing's corner across the face of goal and in.
Dawson had a glorious chance to double his tally eight minutes into the second half when Lescott's header was completely missed by Berahino, but the defender, who took a touch, could only shoot just wide from 12 yards.
