Halifax were disappointing in the first period and only got into their stride as the second half of normal time went on, with Junior Brown bagging an 82nd-minute equaliser.
By the time extra time came around they always looked the more likely to win the game.
Matthew Roney tried his luck early on for Sheffield, but his 20-yarder was tipped over the bar by Craig Ellison.
And a couple of minutes later Roney crossed for Joel Purkiss but he was unable to get on the end of it.
Town, who struggled to hold onto the ball, had to wait until the 14th minute for their first real opportunity after Cavell Coo had cleared downfield.
Stott touched the ball on for Ellis whose left foot cross picked out Kris King, but the unmarked wing back's control let him down.
Town struggled to get into any sort of rhythm, but almost fashioned another opening just past the half hour from a Tony Barras free kick.
He found Coo wide on the right and his cross was headed behind by Lloyd Gelsthorpe under pressure from Stott.
Moments later King almost got in again, but the ball got away from him. He slid in with Sheffield keeper Marc Furness who spilled the ball and while he was still on the floor Ellis curled a shot wide of the target.
Coo defended well from an Edward Cribbley cross and Purkiss should have done better with a header from Jordan Eagers's ball into the area.
At the other end, Jon Smith intercepted Peter Davey's attempted clearance and raced into the area before squaring for Ellis, but he was closed down quickly and the chance went.
Sheffield finished the half with a flourish, but Barras coolly chested back Roney's cross to Ellison, Roney had an effort deflected wide and Ellison took the ball off Greg Wright's head with the defender threatening to nod home.
The second half could hardly have started any worse for Town as Sheffield took a 50th-minute lead when Roney floated a ball in from the left that ended up going inside the far post.
And Ellison had to be on his mettle a few minutes later when he dived to his left to keep out an Eagers shot.
Town came to life around the hour mark with three good chances.
Luke Hinsley had the first - a shot from an Ellis corner that was blocked - before the same fate befell Stott after he got on the end of Paul Gedman's knock down.
Then Baker thumped in a 25-yarder that the keeper somehow managed to keep out, Stott narrowly failing to direct the rebound goalwards.
Ellison's legs denied Roney while Purkiss fired over from close range as Sheffield continued to be a threat.
But the tide was turning in Town's favour and after Baker had fired inches wide and Gedman had headed over the top, they levelled with eight minutes remaining.
Brown played the ball forward and collected a neat return pass from Gedman before lashing the ball home off the upright.
The Shaymen could have won it in 90 minutes, but the keeper saved well from Ellis and Gedman sending the game into extra time.
Town dominated the opening period with Ellis sliding in to put a Brown cross into the net only to be pulled back by an offside flag.
And a goalbound Lincoln Adams header was blocked by a combination of Furness and a defender.
The breakthrough finally came in the 114th minute after Brown had sent Stott away down the right. He did well to work the space for the cross which Ellis volleyed home at the far post.
The game was safe three minutes later when a quickly-taken Baker free kick was hammered home by Stott.

Star Shayman
It is difficult to imagine a Town midfield without Tom Baker who had another impressive game against Sheffield.
He tidied things up when he needed to, was never afraid to launch into a tackle and would have scored but for a great save from Marc Furness in the visitors' goal.
He limped away from a challenge or two which must have caused moments of concern on the bench, but completed the 120 minutes and looked eager for more.
Ashley Stott took his goal tally to three in two games and 14 for the season with a great late strike to seal victory.
He also had a big hand in Town's second which edged them ahead with six minutes remaining as he created the space for the cross that allowed Danny Ellis to fire home.
Luke Hinsley has had few chances to shine since arriving in the summer, but gave an assured display at the back alongside Tony Barras and Lincoln Adams.
He, too, might have scored in the second half but saw a goalbound effort blocked by a scrambling Sheffield defence.
Baker takes the three points on offer in our season-long competition with Stott two and Hinsley one.

Match facts
FC Halifax Town: Ellison, Coo, King (Brown 66), Barras, Adams, Smith (Gedman 46), Gill (Phelan 93), Baker, Ellis, Hinsley, Stott. Subs (not used): Meadowcroft, Kennedy.
Booked: None
Sent-off: None
Goals: Brown 82, Ellis 114, Stott 117

Sheffield FC: Furness, Gelsthorpe, Davey, Wright (Dalby 104), Lipka, Eagers, Cribbley, Hawes (Smith 80), Purkiss, Roney, Outram (Woolley 50).
Booked: Davey (foul 8), Gelsthorpe (foul 90), Eagers (unsporting behaviour 95), Smith (foul 112)
Sent-off: None
Goal: Roney 50

Attendance: 382
Referee: K Haycock

Player points
Junior Brown 19, Jonathan Smith 16, Colin Daniel 10, Peter Moore 10, Ashley Stott 9, Paul Gedman 9, Scott Phelan 8, Tom Baker 8, Jon Kennedy 6, Cavell Coo 5, Adam Morning 4, Lincoln Adams 4, Steve Payne 3, Nigel Jemson 2, Craig Smith 2, Justin Walker 2, Damien Allen 2, Phil Senior 1, Luke Hinsley 2.