The Cumbrians won 4-2 on penalties after looking set to be heading out of the Johnstone's Paint Trophy when they trailed 2-1 with just two minutes to go before veteran Graham Kavanagh gave them a lifeline with his second goal of the game with a close-range header to set up the spot-kick shoot-out.
Adam Collin saved penalties from Wayne Curtis and Mark Duffy to send the Cumbrians through after Garry Hunter and Curtis had twice given Morecambe the lead, only for Kavanagh to equalise on both occasions.
Abbott said: "I am delighted to be into the next round although I thought we could have played better. We were disjointed at times and I was disappointed that some of the lads who I brought into the team didn't really show me enough to be banging on my door for a start at the weekend.
"We started the game terribly and that was disappointing. We were sluggish and were not playing with any real tempo and I had a few words to say to them at half time.
"The second half was a bit better but we made some defensive errors that cost us before we came back and showed some real character and determination to get back into the game."