Walker appeared to have handed Thistle a thoroughly-deserved late win when he conceded an 85th-minute penalty that Aaron Muirhead lashed down the middle of the net.
But he soon fired high into the corner of the goal after bursting to the edge of the Thistle box as Hearts further cut their 15-point deduction for going into administration to an 11-point deficit.
The point rounded off a memorable six days for Hearts fans which started with a derby win over Hibernian and continued with news that they, in the form of the Foundation of Hearts, had been named preferred bidders for the club.
But it was cruel on the home side who were highly impressive in front of a vibrant 6,540 crowd at Firhill.
For the third game running Alan Archibald started with 10 players who won First Division winners' medals last season and they again showed they had fully merited their top-flight chance as they maintained their unbeaten record.
Hearts could not cope with Thistle's first-half movement, energy and passing as the home side pinned their opponents back and created a steady stream of chances.
The visitors pushed men on to the home side's marauding full-backs and had a decent spell after the hour mark but the Jags regained control and Jamie MacDonald saved well from James Craigen, John Baird and Steven Lawless.
Walker bundled Baird over in the box following a short corner to allow Muirhead to net but he sent the visiting fans into wild celebrations moments later.