The match between the league’s bottom two clubs looked to be heading for a draw until Stevenage skipper Cuthbert broke the deadlock when he headed home Charlie Carter’s corner to give the home side the three points that saw them move about Jim Bentley’s side in the table.
The game was always a scrappy one and it was Stevenage who looked the more threatening early in the game with Kelland Watts heading a corner straight at Barry Roche before midfielder Charlie Lakin, fired a long-range effort that flew wide of the post.
Watts then went close again when he produced an acrobatic left-foot volley that was superbly cleared off the line by Alex Kenyon, before striker Kurtis Guthrie let fly with two right-foot efforts that ended up inches wide of Roche’s right-hand post.
Morecambe then had a golden chance to take the lead themselves just after the half-hour when Shaun Miller was played in on goal but saw his effort well saved by home keeper Sacha Bastien.
After the break, Cole Stockton produced a finish of top quality with a volley that beat Bastien low to his left from the edge of the area but Shaun Miller was adjudged offside for interfering with play in what was a controversial decision.
With the game entering the final quarter Stevenage began to forge some openings.
Lakin and Parrett both saw chances fly wide while Sam Lavelle and Steven Old both made vital blocks from shots in and around the box.
Jason Cowley was presented with a great chance but produced a poor ball into the box before Roche was called onto action with a brilliant save to deny Carter with 10 minutes remaining.
Roche could do nothing to keep out Stevenage’s winner two minutes later however when Carter’s right-wing corner was met by Cuthbert whose downward header went through a crowd of bodies on the Shrimps’ line.
Morecambe staged a late rally with Jordan Cranston going agonisingly close to curling in a 95th-minute leveller but Bentley’s side could not find a way through as they dropped to the bottom of the league.
Stevenage: Bastien, Vancooten, Cuthbert, Nugent (Carter35), Denton (Timlin 68), Parrett (Mackail-Smith 88), Lakin, Watts, Guthrie, Cowley. Subs not used: French, Byrom, Sonupe, Iontton.
Morecambe: Roche, Tanner, Lavelle, Old, Conlan (Buxton 83), O’Sullivan Brewitt, Kenyon, Ellison (Cranston 63), Stockton (Wildig 63), Miller. Subs not used: Halstead, Tutte, Sutton, Bakare.
Referee: K Johnson.