Kandol arrowed his header into the top left-hand corner from David Prutton's cross in the 25th minute to secure the win needed to revitalise their League One promotion hopes.
Simon Cox should have pulled Swindon level shortly before half time but headed over unmarked from just three yards out.
United came close to doubling their lead but Swindon goalkeeper Phil Smith tipped over Prutton's corner and Jonny Howson's volley.
It was McAllister's first taste of success in five matches since taking over the Elland Road managerial hot-seat from Dennis Wise and the win reignited Leeds' faltering play-off push.
Leeds had gone seven matches without a win since beating Crewe away in early January and had slipped five points behind the play-off position's.
But Kandol's header ended that barren run. It came from a pinpoint cross-field ball from the influential Prutton, who stamped his mark on the midfield proceedings.
Prutton's cross picked out Kandol, who arrowed a fierce header passed Swindon goalkeeper Smith and into the top right-hand corner.
It was his tenth goal of the season and the one which may prove a turning point in Leeds' season.
Ironically, they had begun badly for McAllister, who had planned to give a debut to on-loan Irish defender Stephen O'Halloran.
But the young Aston Villa wing-back twisted his knee in the pre-match warm-up and was carried off to the treatment room.
But Leeds re-shaped and, after initial Swindon pressure, took a grip of the game with teenager Howson catching the eye alongside Prutton in midfield.
With only one win themselves in six games, Swindon offered little threat, but six minutes before the break they spurned a golden chance of an equaliser.
A corner by Christian Roberts picked out Cox, who inexplicably guided his header over the bar when it appeared easier to score.
The wasted opportunity shook Swindon's fragile confidence and despite mounting pressure, they only tested Leeds goalkeeper Casper Ankergren once in the second half.
That came in the 75th minute when top-scorer Cox found Roberts lurking at the far post, but his rasping shot was brilliantly turned around the post by the Danish keeper.
Five minutes from time, the impressive Howson was just inches off target with a right-foot volley which screamed over the bar. Leeds held on comfortably after that to give McAllister a long overdue champagne moment.