Twice Halifax took the lead in the first half through John Grant and Lewis Killeen before Exeter pulled level with a Billy Jones penalty and an injury-time Steve Flack strike.
But the game hinged on a decision by the referee to send off Denny Ingram for swearing at the linesman for not ruling Flack offside as the players walked off at the break.
And comments to the referee at half-time by Town boss Wilder also resulted in him watching the second half from the stands.
Exeter took full advantage and won the game with two strikes in two minutes midway through the second half. Les Afful scored his first home league goal and Craig Farrell made sure of the result with a brave header.
In the second minute a Wayne Jacobs corner was poorly headed Jon Challinor and when Adam Quinn knocked the ball back across the face of goal Grant made no mistake with a close-range strike.
City equalised in the 11th minute. Challinor was tripped by Matt Doughty inside the box and Jones stepped up to drill the resulting penalty kick under the goalkeeper.
Halifax went ahead for the second time just before the half-hour mark. Danny Forrest found Killeen inside the box and the striker turned and hit an unstoppable strike past Paul Jones from 15 yards.
Exeter drew level in first-half injury time.
Glenn Cronin's fine cross was met by Flack and when his header was parried by the goalkeeper the giant striker thumped home the rebound from close range.
Afful was found by Challinor and the tiny winger skipped past a defender before striking a powerful shot past the goalkeeper from 15 yards to give the Grecians the lead.
The fans were still celebrating when the Grecians struck a fourth two minutes later in classic route one style.
A long clearance by Jones landed on the edge of the box and Farrell leapt superbly to head the ball over the advancing goalkeeper and into the back of the net.