Chris Lines gave the Owls the lead in the 18th minute with a fine strike. The midfielder exchanged passes on the edge of the Shrimps' area with Chris O'Grady before curling a delightful right-footed effort into the corner.
The Owls doubled their advantage with a soft goal after 53 minutes.
Lewis Buxton's right-wing cross caused havoc and, as goalkeeper Barry Roche and centre-back Chris McCready hesitated, O'Grady took advantage with a close-range header that bobbled into the empty net.
The hosts hit back with a penalty just after the hour. Lines was adjudged to have handled Gary McDonald's left-wing cross and Laurence Wilson blasted the spot kick past Richard O'Donnell.
Jim Bentley's charges continued to add pressure, with McDonald testing O'Donnell with a close-range volley and then seeing a late goal disallowed for offside as the Owls hung on for the win.