The on-loan striker opened the scoring with a powerful header from a corner, with Gary O'Neil doubling the lead inside 30 minutes.
Carroll's deft finish put the Hammers clear and, although Graeme Dorrans pulled one back, West Brom finished with 10 men when Youssouf Mulumbu was sent off in stoppage-time.
West Ham were playing on home turf for the first time since the confirmation of their move away from the Boleyn Ground to the Olympic Stadium.
Maintaining their Premier League status between now and August 2016 is of the utmost importance, and three of the five points Sam Allardyce had targeted from their remaining nine games would go a long way to allaying any nagging fears of a swift return to the Championship.
A lively start saw both sides exchange chances before Carroll did open the scoring. Ricardo Vaz Te wasted an opening when he lashed a volley to the right of the post, and West Brom saw a goal rightfully disallowed when Gareth McAuley bundled the ball in at a corner, having collided with Jussi Jaaskelainen in the process.
The home keeper then made a brilliant save that doubled in value shortly afterwards. Romelu Lukaku's free-kick took a deflection that wrong-footed the former Bolton veteran, but he flung out his right hand to touch the ball onto the post and back into his hands.
When Carroll rose above the West Brom defence to head the ball past a helpless Ben Foster, Jaaskelainen's save was put into context, and Mohamed Diame went close to doubling the lead seconds later with a rasping shot that tore past a static Foster to go just over the bar.
Matt Jarvis shot over from close range, but West Ham did add to their lead on 28 minutes through a rapid counter-attack from a West Brom corner. Vaz Te teased the West Brom defence before finding O'Neil, who looped the ball across and beyond Foster with a sublime finish.
The hosts stayed in control until the break, but West Brom were a different proposition afterwards. Lukaku looked more like his old forceful self, but it was defender Billy Jones who went closest to scoring, driving into the box and forcing Jaaskelainen to smother his poked shot just a few yards out.
The arrival of Peter Odemwingie, booed by some West Brom fans for his recent Twitter outburst, upped the Baggies' tempo and the home fans were becoming anxious when Carroll settled the match with a moment of outstanding skill.
James Collins' free-kick dropped over the striker's right shoulder, and as he moved away from Jonas Olsson, Carroll volleyed the ball across and past Foster for his fourth goal in four games.
West Brom pulled one back with two minutes to go from the penalty spot after Lukaku had been caught by Matt Taylor and Winston Reid. Graham Dorrans fired the ball past Jaaskelainen, but the game ended badly for West Brom when Mulumbu lashed the ball in frustration at a West Ham player after conceding a free-kick deep into stoppage-time.
Referee Andre Marriner showed the midfielder a straight red card, meaning he will miss the next three games as West Brom's hopes of European football next season receded further.