The Linnets responded to Saturday's disappointing FA Cup display with a much-improved, professional performance in the replay on Tuesday to set up a second round qualifying trip to Worksop.
Lynn continued to create chances and in the end won at a canter.
The trip had promised to be a proverbial banana skin given Lynn's performance in the first tie, S&L's spirited comeback on that day and a bumpy pitch that threatened to be a great leveller.
But Stewarts and Lloyds' much talked-about pitch turned out to be a bigger threat than the team.
Apart from a second half spell in which they won three corners in a row they were a let-down, and mustered just three shots
Early on Steve Wilson had to save smartly from the hosts' Jamie Gilsenan, but that was as good as it got for the United Counties League minnows
As well as David Staff returning from his ban, the Linnets made three starting changes, Chris Glarvey, Carl Holmes and Malcolm Ndekwe coming into the side.
Two of/this trio were instrumental in opening the scoring on 35 minutes.
Ndekwe's back-header hit the underside of the bar and the back of keeper Shaun Allen before going in the net from a centre by Glarvey.
Before this The Walks outfit had plenty of opportunities, Staff having a free-kick saved and Robbie Harris an effort cleared off the line.
After the goal Ndekwe headed wide and Harris' overhead kick forced a brilliant save from Allen, whose heroics kept the score down.
In the second half, Harris hit the top of the bar from a Staff cross, but he soon. made amends.
Ndekwe and Holmes combined just before the hour mark to carve out a chance for Harris to knock the ball past the keeper and notch his sixth goal of the season - and 10 in just 19 senior games.
Lynn were in this position at The Walks three days previously and yet it ended in a 2-2 draw, but history did not repeat itself.
Holmes put the matter beyond doubt on 65 minutes, set up after Harris had cut in from the left.
As the game went more like a training session for Lynn, sub Chris Bacon had two gilt-edged chances in the final minute, hitting the bar from virtually on the goal line and then producing a fine push around the post from Allen.
Linnets boss Peter Morris said: "A professional performance was required after Saturday and the players obliged on what was a ropey pitch.
"Most of our players were brought up on pitches like that, but we still had to get a result. I was pleased that we hardly gave them a sniff and kept a clean sheet."
Both Morris and director of football Kevin Boon were delighted with the large number of travelling supporters, that made up the majority of the 211 attendance.
Boon said: "The surroundings came as a bit of a shock to the players and supporters, who didn't have to travel after the performance on Saturday.
"But they turned up in force and helped the players get through a difficult night.
"Peter and I can't thank them enough."
It was not to be a night of glory for S&L's Craig McIlwain, whose grandfather had one season at The Walks in 1956 before leaving for Corby. Craig's father Alan was Lynn born and went on to play for Corby, Kettering and Stamford.
This was the fourth time in a row that Lynn had won an away FA Cup replay after drawing at home, the last time being against Cambridge City in 1991.
Lynn: Wilson, Jones, Glarvey, Burrows, Fuff, Defty (Huke 85), Staff, Raynor, Ndekwe (Woodrow 75), Harris (Bacon 75), Holmes. Subs not used: Nobes.
Stewarts & Lloyds: S. Allen, M. McConnell, F Kerr (M. Byrne 70), I. Walker, J. McKay, A. Hall, S. Fan; F Sneddon, M. Curtis (G. Robb 80), C. Mcllwain. J. Gilsenan. Not used: S. McColl, D. Fox.
Referee: Mr M. Blackledge (Peterborough). Attendance: 211. Lynn News Man of the Match - Malcolm Ndekwe