Aaron Wildig's second half header was enough for Derek Adams' side as they held out for a 1-0 win.
It was a game which saw Town move the ball around well at times but failed to really test the Morecambe keeper.
Weaver opted to play the same side which performed well against Forest Green a week ago. It meant Aaron Martin played through the middle in a 4-3-3 formation, with Jack Muldoon and Brendan Kiernan providing support from the wide areas.
A lively opening to the contest saw both sides flirt with chances as George Thomson for Town and Mendes-Gomes for Morecambe both had shots blocked within the first five minutes.
The Morecambe pitch saw a couple Town players lose their footing in the first half and it nearly lead to the Shrimps opening the scoring, as they claimed possession and pushed forward towards James Belshaw’s goal.
It was Ryan Cooney who decided to try test the Town keeper but his curling effort was well covered.
The home side were more than happy to let Town’s centre half have time on the ball, engaging in the press once the ball made its way into midfield. This tactic lead to a sharp counter attack from the Shrimps as the ball was played through to Aaron Wildig who really should have done better with his effort from all of 12 yards out.
The end to the first 15 minutes’ saw Connor Kirby volley over from Brendan Kiernan’s cross, after the winger had done well to keep in a long ball which looked to be gone.
However, despite not having possession in key areas too often, it was Derek Adams’ side who continued to create the better chances.
Stockton skewed an effort wide just before the 20-minute mark, while later in the half, there was a brief scramble in the Town area as a cross skidded around in the six-yard box.
Five minutes before the interval, Wildig once again found himself in a good position to test Belshaw but the keeper wasn’t tested and went into the break without his gloves being worked.
The first shot on target of the afternoon came in the 49th from Morecambe, as Adam Phillips clean strike was well held by Belshaw as the home side looked to kick on.
In the 56th minute, the deadline was broken when Wilding put the ball in the back of the net. It came from a cross towards the back post which made for a simple header for the Shrimps’ number 10.
Morecambe could have had two more goals within the five minutes that followed but Belshaw did really well both times to deny Stockton who was beginning to find space in the Town box.
A frustrated Town dugout saw the visitors up their levels and it produced their best passage of the play so far as Ryan Fallowfield crossed a low ball into Muldoon, who took control of the ball before spinning and shooting just over.
Weaver’s men were now beginning to pass the ball around with purpose and intent but were still lacking that final pass to unlock the Morecambe defence.
With 10 left to play, Weaver opted to bring on Jon Stead for his 50th Town appearance.
However, the striker strugglerd to receive much of the ball and it was at the other end where it looked like the next goal would come as Belshaw was called into action to deny Mendes-Gomes who had broken clear.
Jake Lawlor saw a header saved in the dying minutes but the match ended in defeat for Town.
Town: Belshaw, Fallowfield, Burrell, Lawlor, Hall, Thomson, Falkingham, Kirby, Kiernan, Muldoon, Martin (Stead 80).
Subs not used: Cracknell, Beck, Kerry, Miller, Lokko.
Morecambe: Halstead, Mellor, Lavelle, Davis, Stockton, Wildig, Mednes-Gomes, O’Sullivan, Phillips (Kenyon 82), Cooney, Songo’o.
Subs not used: Turner, Knight-Percival, Diagouraga, Leitch-Smith, McAlinden, Gibson.
Goals: Wildig (56)
Booked: Wildig, Halstead
Referee: Ollie Yates