Taylor struck the winner from 30 yards in what was a contender for goal-of-the-season.
Paul Buckle had earlier given Exeter the lead before Lewis Killeen pulled the visitors level.
The win puts Exeter within a point of the play-offs and within two points of high-flying Halifax.
Exeter had the first shot on goal after 10 minutes when Les Afful blasted the ball wide from 15 yards.
Five minutes later Exeter were ahead. Alex Jeannin's cross from the left was controlled by Afful and Buckle stroked the ball home from close range at the second attempt after his initial effort was blocked by the keeper.
Halifax pulled level against the run of play in the 20th minute. Killeen was first to react to a loose ball on the edge of the box and stuck a powerful low shot into the bottom corner.
Exeter attempted to respond and Sean Devine forced Ian Dunbavin to save well with a shot from a tight angle.
Santos Gaia then headed the ball against the post from the resulting corner.
Exeter were ahead again 10 minutes before half time. Taylor was fully 30 yards out when he stepped up to strike a free-kick but he made no mistake with a superb curling effort into the top corner.
On the stroke of half time Devine headed just over the bar as the Grecians attempted to stretch the lead.
After the break Afful shot over the bar from 12 yards after he was found by Taylor.
Halifax attempted to force themselves back into the game but lacked quality in the final third.
Exeter continued to look the more likely to score and defender Chris Todd has a shot blocked on the line.
But they had to hold on in the last minute as Scott Hiley cleared Adam Quinn's header off the line.