The 33-year-old former Newcastle, West Ham and Liverpool forward has spent the first half of the season at Reading - but his deal at the Royals came to an end earlier in the transfer window.
The club were keen to keep hold of the target man, however Albion have moved to sign Carroll following an injury to January addition Daryl Dyke.
The deal was made official on Friday, three days before the end of the January transfer window, and comes as Albion boss Valerian Ismael faces pressure after just one win in the club's last seven games.
In his pre-match press conference ahead of Albion's key trip to Millwall on Saturday, Ismael said he believes Carroll can have an impact and is hoping his leadership qualities can also aid his side.
He said: “In the short-term it is someone able to help us. To have that impact. We saw in Dike’s first start against Peterborough he played 50 minutes and had five chances to score.
“You see how important it was to get that target man in the box. I think with Andy we have that. He has the experience and that can help us to get the power we need in the opponent's box.
“He is a number nine and this is the reason he is here, to score goals and to have an impact on the opponent’s defensive line.
“He will have an impact on the front three, impact in the opponent's box, link-up, he has desire and leadership.”