After Wayne Rooney had petulantly kicked out at Cardiff midfielder Jordon Mutch after eight minutes, an offence referee Neil Swarbrick deemed worthy of a yellow card when red might have been more appropriate, the England striker duly gave United a 15th-minute lead.
Fraizer Campbell, who struck the crossbar after half-time, levelled matters before David Moyes' men led again as Patrice Evra glanced home Rooney's 45th-minute corner.
But substitute Kim darted in front of the France full-back to produce a similar finish from Peter Whittingham's free-kick in injury time - the midfielder's deliveries having caused United problems all afternoon.
A rapturous Cardiff City Stadium held its breath and then rejoiced as Rooney wasted a late chance to steal the points as Moyes' men finished a frantic weekend of Premier League action in sixth position, seven points behind leaders Arsenal.
Mutch was preferred in the Cardiff midfield to Aron Gunnarsson, with Campbell leading the line against his former club.
Robin van Persie's absence due to a groin injury meant that Javier Hernandez partnered Rooney in attack for United.
Rooney had an effort blocked and Mutch saw his header saved before the two came together in the first-half's early flashpoint.
And the England striker was then, somewhat inevitably, the beneficiary when an error from Cardiff defender Ben Turner resulted in the champions taking a 15th-minute lead.
Turner's loose pass was collected by a rampaging Antonio Valencia and the Ecuador winger's cross was touched on by Hernandez for Rooney to check back and find the net, via a deflection off Gary Medel.
Marouane Fellaini drew a superb save from David Marshall when he headed Adnan Januzaj’s 20th-minute free-kick goalwards, but the young playmaker then erred near his own area, leading to Peter Odemwingie lashing the ball over the crossbar.
Cardiff equalised in the 33rd minute when Whittingham and Mutch combined in midfield and the latter released Campbell for a composed finish.
Campbell, Don Cowie and Whittingham all fired over for the hosts as United laboured in defence of set-pieces, although they demonstrated no such problems attacking them when Evra glanced home Rooney’s left-wing corner on the stroke of the interval.
Rooney's dead-ball deliveries continued to torment Cardiff upon the resumption and Marshall was at full stretch to deny Fellaini before beating away a header from Chris Smalling.
Malky Mackay's team responded once more and Campbell sent an audacious lob against De Gea's crossbar after Tom Cleverley had given away possession, with the rebound narrowly evading Mutch.
United might have earned some breathing space when Rooney headed Valencia’s 59th-minute cross over, while Whittingham sent a first-time effort just wide from distance six minutes later and Cleverley almost diverted a free-kick from the same player into his own net.
Danny Welbeck thrashed wastefully beyond the crossbar 10 minutes from time and the United substitute was made to rue his profligacy when Kim stole in to snatch a share of the spoils, as Cardiff edged three points clear of the relegation zone.
Rooney could still have won it when he latched on to a ball over the top, but his touch was poor.
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