Dutch defender Jos Hooiveld's powered header on the stroke of half-time was enough to earn Saints a 16th successive home victory in front of a record crowd at St Mary's of 32,150.
While delighted with the win, Saints manager Adkins was quick to stress there is still a long way to go before his side can start dreaming of a Premier League return.
Adkins explained: "West Ham were in the Premier League last season, their aspiration is to go straight back up and they are favourites to do that.
"We also have that aspiration and after 12 games of the season we are top of the league and are there on merit.
"We kept a clean sheet, played some good football, created opportunities and scored a good goal from a set play so I am pleased with the three points.
"We have got a lot of belief at the club and are trying to play good football.
"We have beaten a team that will be up there at the end of the season. They are very effective at what they do but we combated that very well.
"But it is only three points and there is a long way to go. All we can do is keep taking it one game at a time."
West Ham started on the front foot, with Matt Taylor testing home goalkeeper Kelvin Davis inside the opening minute only to hobble off soon after Dean Hammond and Richard Chaplow both came close for the hosts before Davis saved well with the legs from Sam Baldock.
That chance seemed to spark Saints into life and moments after Manuel Almunia had kept out David Connolly's volley Celtic loanee Hooiveld nodded in Danny Fox's corner.
Davis made two key stops in the second half to deny Kevin Nolan and, in the dying seconds, Mark Noble - but the visitors were well below par.
Saints were good value for their win, with Chaplow, Rickie Lambert and Guly do Prado all going close to making the victory more comfortable.
West Ham manager Sam Allardyce defended his decision to switch from the 4-4-2 formation employed in Saturday's 4-0 demolition of Blackpool to a more defensive 4-3-3.
He said: "We knew Southampton would be dangerous coming forward and I wanted to nullify their attacking strengths.
"The first half was very tight and we created some chances so were disappointed to concede right before half-time.
"This is not an easy place to come and on another day we would have got something out of it."