The winger managed to find space in a crowded box to convert Ruben Lameiras' free-kick with only seconds remaining.
Simon Dawkins netted twice in a closely-fought encounter with few clear-cut chances.
Dawkins opened the scoring on 19 minutes. Nabil Bentaleb sent Roman Michael-Percil away down the right and the winger zipped to the byline before cutting the ball back for Dawkins, who executed his finish superbly, angling a side-foot into the top corner.
Jack Marriott hit the side-netting for Ipswich but the visitors were soon level as Matt Clarke headed down a corner and Marriott turned home from close range.
Bentaleb broke forward to force Jacob Marsden into a sharp save down to his left but the goalkeeper could do little as Dawkins latched onto Cameron Lancaster's pass and finished via a deflection for 2-1 on 36 minutes.
Luke McGee entered the fray for the second half in place of Jonathan Miles and soon made a fine one-on-one save to deny Marriott. That was matched within a minute by Marsden as Michael-Percil let rip from 25 yards.
Grant Ward was next to test Marsden after a crossfield pass from Giancarlo Gallifuoco and McGee was equal to Teddy Bishop's drive on the hour.
A defensive mix-up allowed Marriott to convert a one-on-one for 2-2 on 78 minutes but typically, the lads pushed forward for a winner, Shaq Coulthirst was inches away from McEvoy's cross on 86 minutes before the late, late winner.
Coulthirst was chopped down on the right, Lameiras whipped in the free-kick and McEvoy probably couldn't believe his luck with the goal at his mercy to secure the victory.
Spurs XI: Miles (McGee, 46); Ward, Ball, Gallifuoco, Stewart (Morne Nel, triallist from club partners SuperSport United, 13; replaced by McEvoy, 62); Bentaleb, Dombaxe, Lameiras; Dawkins, Lancaster (Coulthirst, 62), Michael-Percil.