The 23-year-old England striker has joined the Hammers for the rest of 2012-13 campaign after a summer of intense speculation over his future at Anfield.
Carroll, who moved to Liverpool from Newcastle for a British record £35 million just 19 months ago, has failed to establish himself on Merseyside.
He had been linked with a return to the Magpies but it was his old Newcastle boss Allardyce who acquired his services a day before the closure of the transfer window.
Carroll told West Ham's website, www.whufc.com: "I know the manager well and some of the lads, so it is nice for me to come to a place where I know people.
"I'm hoping to add a lot of goals and create a few chances. I know Sam from Newcastle and he was a massive reason for me coming.
"When I worked with him at Newcastle it was great, so it was a no-brainer really.
"It's great to be here at West Ham and all signed up. I just can't wait to get started now. I want to be playing games and obviously hoping to score some goals."
Carroll scored nine goals for the Reds last season but did not command a regular starting place.
His form improved towards the end of the campaign but new Liverpool manager Brendan Rodgers has conceded the player did not feature in his current plans.
Allardyce said: "I think for all parties this is a great signing.
"Andy is an all-round footballer, but because he is 6ft 3in and one of his strengths is his aerial power everybody dismisses the ability he possesses on the floor.
"Hopefully he can score goals for us and we will be in a very good position at the end of the year in the Barclays Premier League."
Watch: Andy Carroll loaned to West Ham... More Videos
Rodgers has claimed the move does not mean Liverpool are writing off the investment they have made in Carroll.
He said: "I believe it has been difficult for him in his period here.
"He hasn't played anywhere near as regularly as he wanted to and he wanted this to be a season where he was going to be playing regularly.
"Obviously I couldn't guarantee that and obviously getting later into the window it still wasn't the case.
"The club made a monumental investment in Andy but it is no good him being on the bench. At this moment in time there are others ahead of him.
"We are making a strategic decision to let Andy go out and get games and experience."