Pitman crashed over from close range four minutes from time to finally put the seal on the win.
Steven Shingler added the conversion to his four earlier penalties to deny Glasgow a losing bonus and lift Scarlets up to sixth.
But this victory should have been more convincing after Scarlets wasted several scoring chances either side of half-time.
Scarlets were disrupted before the kick-off when Regan King – set to make only his second start of the season – had to withdraw with an injury.
Rhys Priestland, released by Wales, stepped into the fly-half place with Shingler moving into midfield.
Brother Aaron led out the team for his 100th regional appearance.
Glasgow were forced into a front-row re-shuffle after barely 20 minutes as tight-head Zander Fagerson fell foul of referee David Wilkinson.
Coach Gregor Townsend acted decisively, sending on Rossouw de Klerk in place of Fagerson after Glasgow conceded a third penalty for a scrum offence.
Scarlets were further disrupted when captain John Barkley took a blow to the head and had to be replaced in a half which saw Shingler kick Scarlets 9-3 ahead by half-time.
The Warriors, beaten just once in the league before this match, upped their game early in the second half with Niko Matawalu on at scrum-half sparking a series of attacks.
The result was two further penalties for Horne as the visitors seemed poised to take control. But the Scarlets rallied strongly with first Harry Robinson and then Shingler denied by last gasp tackles.
Shingler edged the home team ahead with his fourth penalty and had the simplest of conversion when Scarlets’ late ascendancy was rewarded when substitute Pitman plunged over near the posts for the decisive score.
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