James McClean opened the scoring for the visitors in the 30th minute with his first goal for West Brom in 16 months, ending a solo weaving run with a shot from just outside the area.
Carroll made it 1-1 on the hour mark, using his height to leap above two defenders and head a cross past keeper Ben Foster, breaking his own 14-game goal drought - the striker's longest dry spell in almost a decade.
After an action-packed final half-hour, Carroll sealed the win in the fourth minute of stoppage time, blasting in a shot from a tight angle from a cross by Marko Arnautovic.
West Brom manager Alan Pardew praised his team's hard work, but also noted the impact of having to play 48 hours after their last game, having lost midfielder Matt Phillips to injury in the warmup for Tuesday game.
"It's such a shame, it's heartbreaking for the players to put that effort in and concede with literally the last kick of the game," Pardew told the BBC.
"It's ridiculous to play this game. I've lost Matty Phillips in the warmup, and that was a major blow to us because that meant we had to change our system. So we had a lot of things go wrong for us tonight and most of it is down to the turnaround that we had," he said.
West Bromwich Albion will have until Saturday to recuperate before an FA Cup third-round trip to Exeter.
West Ham must gear themselves up for a trip on Thursday night to in-form London neighbours Tottenham Hotspur.
(Reporting by Hugh Lawson, editing by Ed Osmond)
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