They were left to rue incidents at either end of the pitch as Vadaine Oliver missed a glaring chance with the game goalless, moments before the visitors were awarded what the Shrimps felt was a contentious penalty.
Jim Bentley had made two changes from the Morecambe side humbled by Crewe Alexandra seven days earlier.
Andrew Tutte’s injury ruled him out while Rhys Oates was dropped to the bench.
In came Newcastle United loanee Josef Yarney for his debut, while Aaron Wildig played the support role behind Vadaine Oliver.
A brisk start saw Steve Old head Liam Mandeville’s corner off target, Kevin Ellison volley fresh air and Oliver fire off target from 20 yards.
However, Exeter withstood that early pressure and began to gain a foothold with Nicky Law especially impressive in midfield until injury forced his early exit.
Jonathan Forte curled an effort wide, Hiram Boateng sent a free-kick narrowly off target and two penalty appeals came to nothing.
Jake Taylor saw a shot held by Morecambe keeper Barry Roche before the Shrimps should have taken a 34th minute lead.
Jordan Cranston’s 30-yard shot looked optimistic but it fell to Oliver, unmarked and six yards out, but he could somehow only prod the ball weakly at City keeper Christy Pym.
His generosity haunted the Shrimps as, with half-time approaching, they fell behind.
Lee Holmes outpaced Cranston and progressed into the penalty area, where he went to ground under pressure from the defender.
Referee Thomas Bramall exchanged looks with his assistant Helen Byrne – who offered little in the way of assistance – and a penalty was belatedly awarded.
Sweeney kept his nerve to beat Roche from the spot, even if the keeper managed to go the right way.
As with the first half, the Shrimps had half an opportunity from an early set piece as, on this occasion, Alex Kenyon met Mandeville’s corner but could not direct his header goalwards.
Zak Mills fired over from the corner of the Exeter area but the visitors always looked threatening on the break with Sweeney and fellow full-back Dean Moxey keen to get forward.
So it proved as Exeter doubled their lead on 66 minutes when Sweeney sent over an excellent cross from the right and Stockley converted from close range.
Bentley rang the changes with A-Jay Leitch-Smith, Lamin Jagne and Rhys Oates all brought in during the second half.
Leitch-Smith directed a header at Pym, though he was too far out to seriously test the visiting keeper.
Jordan Tillson’s timely intervention dispossessed Andy Fleming with the midfielder about to shoot, while the lively Oates fired off target after beating two men and cutting inside.
He also fired over under pressure from Sweeney as, despite four minutes of added time, Bentley’s players could only fire blanks.
Morecambe: Roche, Mills, Yarney, Old, Cranston, Fleming, Kenyon (Jagne 78), Mandeville, Wildig (Leitch-Smith 65), Ellison (R Oates 78), Oliver. Subs not used: Halstead, Conlan, Oswell, Hedley.
Exeter City: Pym, Sweeney, Martin, Brown, Moxey, Taylor, Law (Tillson 37), Boateng, Holmes (Martin 72), Stockley, Forte (Abrahams 83). Subs not used: Hamon, Woodman, Jay, J Oates.
Referee: Thomas Bramall.