The home side took the lead with just 80 seconds on the clock. Striker Jack Redshaw received the ball 25 yards out and arrowed in a first-time, right-footed shot that gave Steve Mildenhall no chance as it flew into the bottom right-hand corner.
Redshaw went close to doubling the home side's lead with one effort cleared off the line by Tom Parkes before the visitors hit back early in the second half when Joe Anyinsah crossed from the left and Eliot Richards nipped in between defender and goalkeeper to head past Barry Roche from six yards out.
Both sides then came close to snatching the three points in a pulsating final 20 minutes.
Lewis Alessandra headed a Robbie Threlfall cross against the foot of the post before Richards wasted a glorious chance to double his tally, shooting tamely at Roche when through on goal.
Morecambe had penalty claims rejected deep to stoppage time when Chris Holroyd was bundled over but referee Carl Berry waved play on.