Goals from Lee McEvilly and Paul Mullin, either side of half time, secured the points and helped Accrington improve their league position.
But the home side got out of the blocks slowly and were rocked when Wayne Curtis gave the visitors a surprise 13th minute lead.
Having seen Michael Twiss have a shot saved a minute earlier by Accrington goalkeeper Paul Crichton, the visitors used that as their cue to push forward and break the deadlock.
However, McEvilly slotted home in the 36th minute to level matters and it was Mullin who proved to be the hero with a goal in the 71st minute which left Morecambe wondering what might have been.
Morecambe had more shots than Accrington, but from their 12 efforts only six were on target, going some way to prove that their finishing needs to improve.
McEvilly and Mullin led from the front for Stanley, and they could have had more goals if it had not been for some wayward shooting.
The Shrimps also had their fair share of chances, and Gary Thompson went close with a shot that just flew over the crossbar.
At the other end Steve Jagielka saw a shot blocked as Accrington piled on the pressure, but it was Thompson again who, against the run of play, should have scored but his shot was blocked midway through the first half.
Jagielka went close for Accrington early in the second half, while Garry Hunter fired wide for Morecambe as the game hotted up.
But it was Mullin who swooped to give Stanley the lead for the first time with a sweet header in the 71st minute.
But, however hard Morecambe tried, they could find no way through Accrington's defence and the home side deserved the spoils.