The highly-rated full-back has made the step up into the Premier League from Ipswich Town for a fee believed to be in the region of £3.75million, but which could rise to £7m if certain clauses are met.
Allardyce considers Cresswell to be a shrewd acquisition, with several top flight clubs having shot admiring glances in his direction over recent years.
West Ham believe they have pulled off quite a coup in landing a player who, at 24 years of age, will have a big part to play in their present and future.
Allardyce told the club’s official website on being able to do a deal for a talented Englishman at a reasonable price: “Aaron is a very interesting case because, as far as we looked at other targets, when I looked at his career history, spoke to Mick McCarthy and looked at what he has done in the last couple of years, he was right on the doorstep.
“He started at 17 at Tranmere Rovers and played 70 games and Paul Jewell took him to Ipswich when he became manager there. He has played nearly every game over the last three seasons, so he never misses any games through injury.
“He is 24 years old and there has been a very good, steady development over those years. He has over 200 games in the bag.
“He was in the PFA Team of the Year last season, so all of the players he played against saw him as the best left-back in the Championship.
“Of course, the price was right, which was the biggest factor in our transfer dealings this season. If the player is right and the price is right, we do the deal.
“The biggest stumbling block is the size of the fees that are being asked for today. This one was a very, very good signing indeed, not just for right now but also for the future.”