Goals from Greg Pearson, Elliott Ward and Phil Lumsden secured the three points for Tony Carr's team, who netted all three strikes within the first half an hour.
After the hosts had caused a scare in the opening two minutes with a shot that struck the post, Hammers took the lead after just six minutes, when a through-ball from Billy Mehmet released Pearson, who finished well from just inside the penalty area.
Nine minutes later, the visitors doubled their lead, when James Allen's low free-kick from the right was flicked home on the volley by Ward.
Allen was the provider again in the 28th minute, squaring the ball across the box for Lumsden to slot home the third.
Surprisingly, that completed the scoring, as Hammers eased off slightly in the second half and kept possession well to further frustrate the Saints.
Pearson went close again, while Daryl McMahon was denied by a fine save from the opposing goalkeeper, while at the other end Hammers keeper Glenn Jackson was only tested once.
Said Tony Carr afterwards: "We played extremely well in the first half, took our chances, and to be honest the game was safe once we got the third goal.
"We didn't play as well in the second half, but the boys were always in control and never looked like getting into any danger, so I have no complaints."
Team: Jackson, McClenahan, Tattam, Johnson, Ward, Lumsden, Allen (Akinsete 70), McMahon, Mehmet (Carrick 70), Pearson, Cole (Eastwood 75). Unused subs: Khan, Fletcher.
Meanwhile, Kevin Keen's under-17 were unlucky to lose at Coventry City, where a 2-0 scoreline didn't reflect the true story of the match.
After conceding a poor goal direct from a corner in the first 10 minutes, Hammers were by far the better side for the remainder of the game but failed to find the breakthrough and the hosts added a second in the dying moments.
Said Kevin: "We played very well, in fact it was probably the best 90 minutes we have put in so far this season, but sadly it just wasn't our day.
"The players worked very hard and were on top for the majority of the game, but we conceded a bad goal early on and couldn't make our mark at the other end. Darren Blewitt had a snap-shot from a corner that was somehow kept out by an outstanding save from Coventry's goalkeeper and we just didn't have the rub of the green.
"They did everything I asked of them, apart from score, and I am proud of their performance. If we carry on like that, it will be a successful season."
Team: Reed, Wright (Welsh 85), Parrington, Laws, Blewitt, Tucker (Singh 40), Bunce, Noble, Cohen, Stokes (Collington 75), Yao. Unused subs: Wayman, Henry.