Head coach Mel, who was appointed on January 9, opted to hand Saido Berahino a first start in attack since February and was rewarded when the England Under-21 international struck in the 11th minute, settling Albion's early nerves.
Andy Carroll went close for West Ham either side of Berahino's early strike and almost made West Brom pay for missed chances after the interval when he rattled the crossbar from a 71st-minute corner.
Carroll kept the ball alive well from a sixth-minute corner to draw a sharp near-post save from Ben Foster and Jonas Olsson followed his goalkeeper's heroics by producing a last-ditch challenge to deny the West Ham centre-forward.
The impact on West Brom's overall performance was initially negligible and Foster had to be at his best once more to deny Carroll, who got across Steven Reid to head Mohamed Diame's inviting cross goalwards.
Reid's fellow full-back Billy Jones was inches away from doubling the hosts' advantage with a flying header from Amalfitano's whipped 25th-minute delivery.
Immediately after Berahino's touch had deserted him in the 62nd minute, Sessegnon teed up his fellow forward to fire straight at Adrian from close range and pass up a glorious opportunity.
Berahino's predatory instincts were then lacking as he failed to turn home Youssuf Mulumbu's speculative effort.
Carroll almost made him pay when he was inexplicably left alone to strike the bar from Matt Jarvis' delivery, while Sessegnon scuffed a 75th-minute volley and Adrian denied him the goal his efforts deserved.
