The Cumbrians have never appeared in a Football League play-off but an unbeaten March campaign has bolstered their chances.
Stockton broke the deadlock after 11 minutes with his sixth goal since joining on loan from Burton in the January transfer window.
The former Morecambe goal machine doubled the Bluebirds’ lead in the 22nd minute with a simple tap-in after Dom Telford’s shot following a mazy dribble was stopped on the line.
Barrow looked set for a routine three points but Grimsby kept up the pressure in the second half and saw Kieran Green hit the post with a header.
However, Pete Wild’s side sealed victory after 81 minutes as midfielder Robbie Gotts drilled in a low shot from the edge of the area.
Gotts’ goal was extra insurance as Justin Obikwu fired in after 88 minutes to give the Mariners a consolation.
Barrow’s latest win stretched their unbeaten run to six games – all in March – with four victories and two draws and leaves them fifth in League Two.
Artell’s side had the better of a closely-fought first half, with Donovan Wilson rattling Wimbledon’s crossbar.
Both sides struggled to carve out a clear-cut chance during a cagey second half, and Grimsby held on to secure back-to-back clean sheets in League Two.
The result moved the Mariners seven points clear of the relegation zone, while the Dons are now three points off the play-off places.
“We know we are hard to beat, it was a hard-fought point,” he said.
“In the second half Wimbledon asked questions of us but that is back-to-back clean sheets for the first time in, I don’t even know how long.
“I thought we were the better team in the first half, we hit the bar, we had a great penalty shout turned down, Harvey Rodgers has had a great chance from a free-kick and we looked the better team. It was a really good point in the end.
“I thought the referee missed a couple of big decisions if I am being honest but that is the way it goes when you are away from home, that is what can happen.
“We should have scored while we were on top, that’s the next stage, we made it hard for ourselves in certain respects in the last 15 minutes.”
Wimbledon boss Johnnie Jackson was happy with a point in the wake of Saturday’s memorable 1-0 win over MK Dons.
“I think it took a lot out of all of us, it was a monumental occasion and to have to go again three days later, that is difficult,” he added.
“To put the effort in that they did on the back of a huge effort on the weekend and get four points from two games in three days, it is not bad.
“I think on the balance of the game the result was about fair, it was close but it is another point on the board.
“We wanted to win and the plan was to get the three but you can also lose these games and our opponents are fighting for their lives so credit to them for making our lives difficult.
“I don’t think we were at our best and when you are not at your best it is important you don’t lose.
“It is going to be a real fight, you have seen today there are no easy games, the team that we beat here on Saturday has gone and won at the league leaders.
“This is League Two for you, they were never going to come here and roll over, they were always going to put up a fight.”
Pep Guardiola's top-flight champions eased past Bristol City 3-0 on Tuesday, while United backed up their EFL Cup success with a 3-1 comeback victory over West Ham.
City will face former defender Kompany, who lifted four Premier League titles at Etihad Stadium, after being drawn at home to runaway Championship leaders Burnley following their late 1-0 win against Fleetwood Town on Wednesday.
United ended a six-year trophy drought on Sunday by defeating Newcastle United and will continue their quest for further silverware when they host Fulham at Old Trafford.
Fourth-tier Grimsby Town remain the shock package in this season's FA Cup after Premier League Southampton fell to a 2-1 home loss and became the Mariners' fifth victim from a higher division in the 2022-23 campaign.
The League Two side's reward will be a trip to in-form Brighton and Hove Albion, who edged past Stoke City 1-0 in the fifth round on Tuesday.
Sheffield United were another surprise winner in the last 16 of the competition, defeating Tottenham 1-0, and will host Blackburn Rovers, which ensures at least one Championship side will reach the semi-finals.
All ties are set to be played between March 17 and 19 with the quarter-final contestants one win from a trip to Wembley.
FA Cup quarter-final draw in full:
Manchester City v Burnley
Manchester United v Fulham
Brighton and Hove Albion v Grimsby Town
Sheffield United v Blackburn Rovers
