Boro started superbly and found themselves two goals ahead in the first eight minutes.
They took the lead after just five minutes when a low cross from John Mousinho was met by Scott Laird, who nipped in between two Morecambe defenders to slot the ball past Barry Roche from close range.
Stevenage doubled their lead three minutes later when Laird provided the cross from the left and Morecambe defender Kieran Charnock slashed the ball into his own goal from eight yards out.
Stevenage continued to pile on the pressure with Charlie Griffin wasting a glorious chance after 15 minutes and Lawrie Wilson forcing Roche into a decent save with a left-foot shot.
After the break Shrimps substitute Stewart Hendrie forced Chris Day into a save with a shot from the edge of the area and Stewart Drummond volleyed an Andy Parrish cross inches wide late on, but it was too little, too late.
Source: PA
Source: PA