Matt Richards, Terry Gornell and substitute Byron Harrison scored the goals, while Richards also saw his penalty saved.
Scott Brown made two fine stops to deny Padraig Amond and Ryan Williams inside the first 10 minutes as the Robins lived dangerously. But they took the lead in the first minute of time added on at the end of the first half.
Gornell was tripped by Mark Hughes and Richards' free-kick from 30 yards out evaded everyone and curled inside the left-hand post.
Goalkeeper Barry Roche was unsighted by a crowd of players in front of him and Richards celebrated his fourth goal of the campaign.
Things worsened for Morecambe in the second minute of the second period when skipper Hughes scythed down Sam Deering, earning him a second yellow card and Jim Bentley's team were down to 10 men.
Cheltenham had a great chance to make it 2-0 in the 53rd minute when Sido Jombati's cross was handled by Robbie Threlfall and a penalty was given. Richards stepped up, but Roche dived low to his right to keep it out.
However, Gornell added the second in the 62nd minute with a determined run and fine finish inside the top right-hand corner of the net before Harrison made it 3-0 in the 82nd minute with a crisp finish from inside the area.