Seymour grabbed his try with five minutes left on the clock and just three minutes after Wales fly-half Dan Biggar appeared to have stolen the win for the Ospreys with a calm dropped goal. Duncan Weir's conversion attempt rattled off a post, giving Ospreys the hope that they could still swing the result with another score.
Ospreys churned through the phases to set a position near the Glasgow 22. Again, Biggar dropped back into the pocket, but this time Glasgow were alive to the danger. Niko Matawalu charged the kick down and raced off after the ball, a series of hacking kicks propelling it to the other end of the field.
In the end, the Ospreys defence got back to cover the danger, but time had run out for the Welsh side and referee John Lacey's full-time whistle gave Glasgow the victory they craved in a game coach Gregor Townsend had billed as the biggest of their season.
As Finn Russell had also had a late touchdown chalked off for a knock-on, there was little question that Glasgow's late surge made their victory a deserved one. Yet they laboured their way through too much of the game, pasted for territory in the first half and then undermining their own cause with nervous handling errors later in the game.
Still, the result leaves them just one point behind the Ospreys in the PRO12 table, but in a far stronger position than the Swansea side on the strength of having two games in hand.
In temperatures more typical of mid-January than late March, Glasgow started with the disadvantage of playing into a tugging wind that gave their kickers a devil of a job judging length and line. That factor clearly helped Ospreys in the first quarter, and Glasgow could consider themselves a little lucky to concede just three points in that period, those provided by a Biggar penalty in the 19th minute.
Against that, the Warriors had done precious little going forward, and they could thank their dominant scrum for keeping them so close on the scoreboard. Finn Russell did make one lovely break shortly after Biggar's opening score, but the move petered out when the ball was turned over in the Ospreys' 22.
Glasgow seemed to be in strong danger of falling further behind when Peter Murchie needlessly took a quick throw to Matawalu deep in his own half, only for the Fijian to drop the ball and give the Ospreys set-piece possession. Yet again, though, the Glasgow scrum saved the day, the Welsh side conceding a penalty as they caved in under the power of the home pack.
It would be easy to fault Glasgow for trying to do too much, too deep, but the wind made kicking for safety a perilous tactic. That said, there was a scrappiness about much of what the Warriors did, and they never really found any sort of rhythm during the first half. All things considered, they could be pretty satisfied to reach the interval with just that three-point margin against them.
As expected, Glasgow began to turn the screws in the third quarter, putting the wind that had tortured them for the first 40 minutes to good use. Yet if they were trying to imitate Ospreys' first-half tactics, they came up short as they failed to match their opponents' earlier composure on the ball. Ospreys certainly took more territory off Glasgow in the second period than Glasgow had taken off Ospreys in the first.
Still, Glasgow did win the positions that allowed Weir to clip over penalties in the 48th and 55th minutes. The Warriors scrum also continued to dominate, and players like Rob Harley and Jonny Gray put in some magnificent work in contact, never allowing the Ospreys time to settle into a groove.
But just when it seemed that Glasgow were about to pull away with it, the Ospreys came thundering back into the contest. After churning through their phases, they won a penalty in the shadow of the posts, and Biggar hammered the ball over to draw his side level with 15 minutes left on the clock.
Glasgow still looked the better side, but their composure continued to let them down. Shortly after Biggar's equalising penalty they put in a powerful forward drive but then turned possession over when they failed to get the ball on the floor. In a season of many frustrations, the Glasgow crowd were growing restless.
Their fears were justified when Biggar landed his dropped goal. But Glasgow held their nerve and a clever kick by Matawalu allowed Seymour to race away and grab his winning try, beating Biggar in the chase to the touchdown.
It was a lovely piece of skill on an evening when such moments were all too rare. But as golfers always say, it's not how, but how many, that counts.