Terry McPhillips' side were dominant throughout and on this occasion got the goals to match the performance.
A bright start from the home side was eventually rewarded on 28 minutes when following Danny Laverty's deep cross from the right, Modou Cham delivered back across from the left to where Antonie Boland arrived at the back post to fire home.
Rovers added a crucial second goal just before the break. Bradley Ware intercepting to prompt a counter-attack from Dean Rittenberg, the attacker saw his effort saved, but Jordan Preston was on hand to follow in on the rebound.
Having scored three minutes before the break, Rovers then scored three minutes afterwards to make it 3-0. Good work by Bradley Bauress saw him feed Brice Wassi and the midfielder turned and passed to Bradley Ware who thumped the ball into the back of the net emphatically from 25-yards.
On 69 Rovers added a fourth goal to confirm a fine afternoon. Central defender Ryan Thompson intercepting in his own half before racing down the left, his centre was met by Modou Cham who headed into the back of the net powerfully from distance.
Two minutes later West Ham pulled a goal back, a cross from the left was met well by the fast arriving attacker but it was mere consolation.
Rovers had further chances to stretch their advantage in the closing stages but failed to add to their four goals.
Speaking after the match, coach Terry McPhillips said: "We were miles better this week, we had a good go, we played hard and competed well. It was much improved from the Reading game. So I'm pleased with the lads, we've got to keep it going now.
"I thought Brice Wassi was the man of the match for us, i thought he had his best game of the season. It was a good team performance.
"We'll now focus all our efforts on next Tuesday's FA Youth Cup tie against Leicester City at Ewood Park."
ROVERS: Grant, Laverty (sub Humphreys 66 mins), Butterwick, Thompson, Bauress, Boland, Ware, Wassi, Cham, Rittenberg (sub Pero 60 mins), Preston (sub Cotton 60 mins).
Subs not used: Raya, Langford.