SPL: Partick Thistle 1 Hearts 1
Aaron Muirhead gave the home side a deserved lead from the penalty spot in the 86th minute, but 20-year-old winger Walker's superb equaliser two minutes later gave Gary Locke's men a share of the spoils.
Hearts are rooted to the bottom of the Scottish Premiership after being deducted 15 points for entering administration, but took another step towards the zero-point mark by following up their derby win over Hibernian with a draw at Firhill.
Newly promoted Partick have started their first season in the top flight for nine years strongly, taking four points from their opening two games.
And Alan Archibald's men looked full of confidence from the outset on Friday as they dominated the early exchanges. Striker Kris Doolan had the ball in the net after just nine minutes, but his effort was ruled out for offside.
Conrad Balatoni then wasted a good chance for the hosts when he headed Stephen O'Donnell's cross over in the 20th minute, before captain Sean Welsh hit a rasping shot just off target 10 minutes before half-time.
Despite Partick being well on top, the sides went in level at the interval and the visitors improved after the break.
However, the home side were still in control and finally edged ahead with just four minutes left. Substitute John Baird was brought down in the area by Walker and Muirhead emphatically smashed the resulting penalty past Jamie MacDonald.
But Hearts refused to give up and Walker made amends for giving away the spot kick by rifling a 20-yard piledriver beyond Scott Fox after collecting a pass from Jamie Hamill.