Prolific Robins striker Simon Cox had done his best to spoil the promotion party, but Keates made sure Posh waved goodbye to League One by picking up a point.
The fine finishing of hot-shot Cox - in what is rumoured to be his last game for Swindon - and some inspired goalkeeping from Phil Smith were the major factors why Posh were reliant on Keates' late strike.
Posh defender Charlie Lee collected no fewer than four player-of-the-season awards before kick-off - and then promptly laid on the opening goal for the visitors!
Lee's misplaced pass was seized upon by Cox who coolly fired past keeper Joe Lewis in the 17th minute.
Lee twice went close at the other end before Posh hauled themselves level with six minutes of the opening period remaining.
Leading scorer Craig Mackail-Smith steered home his 26th goal of the season from close range after being picked out by a right-wing George Boyd delivery.
But parity was short-lived as Cox pounced again in the 42nd minute to fire home an impressive half-volley.
It was one-way traffic as Posh poured forward to no avail after the break.
Boyd saw strong penalty appeals turned down before convincing referee Mark Haywood to point to the spot moments later after being tripped by substitute Kevin Amankwaah.
But striker Aaron McLean was found wanting from 12 yards as Smith saved his spot-kick - and the Swindon shot-stopper soon took centre stage with a succession of super stops.
He denied McLean, Dominic Green and substitute Shaun Batt while Russell Martin and Gaby Zakuani had efforts cleared off the line before Keates' last-gasp leveller.