Keates left-foot shot, from the edge of the area into the top corner of the net, left Walsall as League Two Champions and Swindon will join them in League One next season after finishing in the third and final automatic place.
Walsall also benefited from Hartlepool's failure at home to Bristol Rovers, and for much of the match at the County Ground Swindon were the better side.
Walsall did not muster a shot on target until the 86th minute when a brilliant diving save from Phil Smith prevented Martin Butler scoring.
Swindon should have been firmly ahead by the time Keates finally equalised to send the three thousand travelling Walsall supporters into ecstasy.
Jerel Ifil scored Swindon's goal in the 52nd minute with his first in the league in a career spanning 149 games, heading home a corner taken on the left by Sofiane Zaaboub.
Lee Peacock hit a Walsall post with a header and sent another effort narrowly over the bar, while Christian Roberts and Blair Sturrock both went close to adding further goals.
But Swindon paid a heavy price for the inability to take chances when Keates responded with his late goal.
Walsall Manager Richard Money was sent to the stands from the dug-out by referee Mike Jones for disputing a throw-in near the halfway line during the second half. But Money was quickly down onto the pitch to join in the celebrations with his players at the final whistle.
Meanwhile Swindon's supporters in the largest crowd of the season, 14,731 were also jumping for joy at the final whistle for clinching third position.