The only goal of a first half low on chances was a 37th-minute penalty from Gary McSheffrey. The Sky Blues striker coolly converted the spot-kick after he was brought down in the process of rounding Morecambe goalkeeper Barry Roche.
The League One side were on top after that at the Ricoh Arena and Richard Wood almost added a second goal before the break but he headed over.
However, it took just 17 seconds of the second half for the hosts to double their lead. Carl Baker curled a left-foot shot into the top corner of the net from 20 yards out following good work by McSheffrey and Franck Moussa.
Wood thought he had made it 3-0 after 67 minutes when he pounced on a rebound from close range but it was ruled out for offside.
Kevin Ellison pulled a goal back for the Shrimpers 10 minutes later with a goal similar to Baker's. Ellison reacted quickest to a poor clearance to shape a right-foot shot into the top corner of the net from 25 yards, but the 1987 cup winners held on.
