The 18-year-old was the Shrimps' hero with a goal in either half to send them through to the second round and add further misery to a Grimsby Town side still looking for their first win since March.
Taylor put Morecambe ahead right on half0time, connecting brilliantly with a 25-yard free-kick from Craig Stanley to head past Grimsby keeper, Phil Barnes.
And the teenager added his second after 77 minutes when Barnes made a hash of a clearance to present the ball to Dairmuid O'Carroll, who hit it first-time across the face of the goal for Taylor to sidefoot home from close range.
A goal from Mariners' defender Robbie Stockdale two minutes from time gave the visitors a glimmer of hope, but despite some intense pressure Morecambe held on to move into Sunday's draw.
Morecambe had the better of the opening stages with Henry McStay heading over an Adam Yates cross after 15 minutes, but two minutes later Barry Roche had to save with his legs to block a firm shot from Nathan Jarman.
Ryan McCann replaced Yates after 21 minutes after the right-back limped off with yet another injury for Sammy McIlroy to add to his long list.
And McCann was involved almost immediately by playing in Taylor, but the striker's effort was well blocked by Barnes.
Moments later McCann was involved again with a cross which Craig Stanley diverted against the post with Barnes beaten and the same player provided another dangerous ball into the area which Stewart Drummond headed over.
Just as it seemed Grimsby would hold on to the break Morecambe struck. Chris Llewellyn gave away a needless free-kick on the right and Stanley's superb delivery was met by Taylor.
Grimsby raised their game at the start of the second period with the on-loan Adam Proudlock volleying inches over, before smashing a half-volley against the crossbar which beat Barry Roche.
The Morecambe keeper then got down well to save from Jean Paul Kalala who then produced a fierce 25-yard volley that whistled past Roche's post.
After soaking up the pressure Morecambe began to get the upper hand again. O'Carroll went close hitting a low drive from the edge of the box that was deflected just wide before the same player turned provider for Taylor to tap home his second after a rare error of judgement from Barnes.
Stockdale pulled one back for Grimsby with two minutes to go, but they could not find their way through the Morecambe defence again as the Shrimps celebrated their passage into the next round.