City were forced to endure a torrid opening spell of pressure from the home side but they held on and scored a breakaway goal of their own which gave Iffy Onuora's side the platform they needed to seal the three points.
Matty Blinkhorn should have given Morecambe the lead in only the second minute after a mistake by Imps keeper Ayden Duffy.
The Lincoln shot-stopper let the ball slip from his grasp on the edge of the area and presented Blinkhorn with an open goal but he missed the target with the ball clearing the crossbar by inches.
The Shrimps' pressure continued with Blinkhorn finding space in the box again to get behind Lincoln skipper Lee Beevers but dragged a right-foot shot inches wide of the right-hand post.
Carl Baker then came close with an audacious lob from wide on the right which had Duffy back-pedalling as the ball just beat the crossbar.
Blinkhorn came close again with a curler on 24 minutes and was then denied by Adie Moses who cleared off the line after a neat shot on the turn from the Morecambe striker.
But then, against the run of play, Lincoln scored. Gary King broke free in midfield and found Wright unmarked on the left-hand side and he slotted home comfortably past Scott Davies to register his 14th goal of the season.
Duffy produced an excellent save from Garry Thompson in first-half stoppage time and the pressure continued after the break.
Craig Stanley saw an effort cleared off the line by Gary Croft on 53 minutes and Morecambe's hunt for a goal continued as Blinkhorn hit the post from the edge of the box with an excellent curler.
To make matters worse for the home side Lincoln hit a second 20 minutes from time when full-back Green cut inside from the right and produced an excellent left-foot finish past the stranded Davies.
Louis Dodds was denied a third when he hit the post from the edge of the area before Morecambe scored the goal they deserved when Blinkhorn struck three minutes into stoppage time.