The Shaymen looked down and out when Adi Yussuf and Nathan Blissett had put Solihull 2-0 up just before the hour-mark.
But the introduction of Cameron King, who set-up Town’s first goal, helped to turn the tie, with Kosylo producing a wonderful free-kick to deservedly rescue a draw.
The sides will meet again at Solihull on Tuesday night for a place in the third round of the Trophy.
Town got better as each-half went on, with Solihull quicker and stronger in the early stages, winning the ball too regularly and finding it too easy to get within reach of goal.
Influential midfielder Darren Carter had a couple of half-chances early on, while the ball wasn’t sticking up-front for Town, who weren’t helping themselves with some poor decision-making when attacking, with Scott Quigley not opting to shoot when well-placed after Jordan Preston’s shot was blocked, and then losing the ball.
Carter’s rasping drive then arrowed just over from 30 yards, but The Shaymen then began to play much more of a part in the contest.
Niall Maher spontaneously burst forward from inside his own half around the half-hour mark, and although nothing came of it, the midfielder’s 40 yard run displayed the kind of adventurous spirit the Town fans wanted to see.
Quigley again chose the wrong option when he ignored Kosylo in space to his right, and ended up turning back and laying the ball off to Southwell, whose shot was deflected behind.
From the corner, Matty Brown’s downward header was cleared away in-front of the goal line by Luke Maxwell.
Chances were becoming more frequent as the half went on, in a much more watchable game than Braintree a week ago, with Town upping their level of performance.
Preston spurned a great chance for Town when he fired over from around the penalty spot when Brown headed a corner back across goal.
Solihull still carried a threat though, with Maxwell’s close-range header turned behind by Sam Johnson, before Yussuf raced onto a loose ball and sent a low shot bobbling across goal and just wide.
Town were missing the drive and energy of Ben Tomlinson and Jacob Hanson down their right flank, but James Berrett made an encouraging first appearance since the 1-1 draw against Chesterfield on October 13, offering composure and awareness in the centre of midfield.
Any momentum Halifax took into the second-half evaporated when Carter’s dinked pass into the six-yard box left Yussuf with a simple finish, but it had been too easy for the visitors to find a route to goal.
And from then until Solihull’s second goal 10 minutes later, there was no suggestion that The Shaymen would produce a response.
So when Nathan Blissett slid in at the far post to convert an excellent low cross from the right by Matty Stenson, it looked a long way back.
Even after that, Solihull looked the likelier to score, with Maxwell and Terry Hawkridge having decent chances.
But out of nothing, substitute King, on his first appearance since December 1, made an instant impact as he released Southwell with a perfectly-weighted through ball, and the striker dinked the ball over the onrushing Ryan Boot.
Town were at least showing some fight now, helped by the introduction of King, who linked play and picked passes effectively, and had some momentum behind them.
Fellow substitute Jonathan Edwards’ jinking run earned Town a free-kick 25 yards out, from which Kosylo curled a delightful effort into the top corner to complete a comeback which, at 2-0, didn’t look on the cards.
Halifax could have won it late on but didn’t take advantage of Boot’s poor clearance, with Edwards caught offside when he would have been clean through on goal.
Halifax: Johnson, Duckworth, Clarke, Brown, Skarz, Berrett, Maher, Kosylo, Preston (King 63), Southwell, Quigley (Edwards 78). Subs not used: Rowley, Hanson, Ferry.
Scorers: Southwell (64), Kosylo (82)
Shots on target: 4
Shots off target: 4
Solihull: Boot, Williams, Flowers, Carline, Gudger, Carter, Maxwell (Osborne 71), Hawkridge, Yussuf (Hylton 90), Blissett, Stenson (Wright 74). Subs not used: Reckord, Daly.
Scorers: Yussuf (47), Blissett (58)
Shots on target: 7
Shots off target: 9
Referee: Andrew Miller
Town man of the match: Cameron King