The Southampton loanee volleyed in Ian Maatsen's centre to open the scoring just a few minutes into the second half but Vincent Kompany's side showed their vulnerabilities late on once again, with Robinson heading past a helpless Arijanet Muric to earn the hosts a point in Mark Hudson's first game in charge.
The point leaves Burnley in the fourth position after 11 games in the Championship season.
Here are the player ratings.
2. Arijanet Muric, 6.5
Often unnervingly calm in possession, the Kosovo international mopped up well from Cardiff's balls over the top. He also held Ryan Wintle's tame effort and palmed away Mahlon Romeo's fierce drive, but could do nothing about Callum Robinson's late equaliser. Photo: CameraSport -Alex Dodd
3. Connor Roberts, 6
The former Swansea City man took a number of late challenges amid the home crowd's boos but kept his head well and helped to shackle the dangerous Callum O'Dowda. Saw a lot more of the ball when he drifted into a more central position in possession but struggled to get forward. Bested by Robinson at the back post for Cardiff's late equaliser. Photo: CameraSport - Dave Howarth
4. Charlie Taylor, 6.5
The 29-year-old was the most experienced man in Burney's backline and he showed all of his know-how when cutting out dangerous crosses from Mahlon Romeo and O'Dowda in the first half. That theme continued in the second 45, with the former Leeds man consistently in the right place at the right time. Photo: CameraSport - Paul Greenwood