On a night when the visitors, marshalled by the brilliant Dusan Tadic, were quicker in thought and deed for long periods, Milinkovic-Savic, who had earlier headed his side in front, thumped an attempted 86th-minute clearance straight at team-mate Nikola Milenkovic and saw it end up in the back of his own net to ensure the game ended 1-1.
Ireland's qualification hopes, which were hanging by a thread before kick-off, look no more realistic with Group A leaders Portugal 11 points better off and Serbia nine clear with just three games to play - but for manager Stephen Kenny, there was at least a measure of respite after Saturday's disappointing home draw with Azerbaijan.
His critics are unlikely to be pacified by a performance which relied heavily on the traditional Irish qualities of hard work and commitment, but the heroics of 19-year-old keeper Gavin Bazunu, who spoiled Aleksandar Mitrovic's record of scoring in every qualifier to date with a string of fine saves, and an impressive full debut by fellow teenager Andrew Omobamidele were significant pluses.
Kenny has now taken charge of 12 competitive fixtures without a victory and the Republic have not collected three points in 16 attempts in total dating back to June 2019.
Opposite number Dragan Stojkovic signalled his intent from the moment he submitted a teamsheet which featured Mitrovic and Dusan Vlahovic in a frontline supported by Tadic and wide men Filip Kostic and Filip Djuricic.
They forced two early corners and were almost gifted the opening goal within the first five minutes by Shane Duffy's miscontrol, although Bazunu came to his captain's rescue by blocking Vlahovic's attempt.
Bazunu had to field Nemanja Gudelj's 14th-minute strike from distance after Duffy had cleared the latest of a series of high balls, but the keeper was tested further by the same man moments later after the midfielder had been allowed time and space to work his way into position.
The visitors took the lead with 20 minutes gone when Milinkovic-Savic got across his marker at the near post to glance Tadic's driven corner past Bazunu despite the keeper getting a hand to the ball.
Rajkovic was relieved to claim Jeff Hendrick's 26th-minute free-kick with the minimum of fuss after Jamie McGrath's progress had been halted illegally by Djuricic. Adam Idah was cautioned, perhaps harshly, for a high boot as he tried to beat defender Strahinja Pavlovic to James McClean's teasing cross as Ireland responded.
Bazunu did well to beat away Mitrovic's 39th-minute strike from Vlahovic's adept lay-off and then dealt with the resulting corner, but Serbia's movement and intricate passing game was repeatedly exposing holes in the home defence.
Having left lone striker Idah painfully unsupported for much of the first half, Ireland attempted to remedy that in the early stages of the second, although Alan Browne was unable to direct McClean's 49th-minute cross at goal amid an encouraging start.
However, Bazunu had to repel Mitrovic's 57th-minute effort after Tadic and Djuricic had combined delightfully and then pull off an even better stop three minutes later to deny the Fulham striker when he had connected with Kostic's cross.
The teenager came to the rescue once again with a double save from substitute Nemanja Radonjic and another from Kostic, having needed McClean's help after losing his bearings 13 minutes from time with the game opening up, and his heroics were rewarded at the death.
In his efforts to clear substitute Callum Robinson's cross, Milinkovic-Savic blasted the ball at Milenkovic and looked on in horror as the ball ricocheted into his own net to spark a late onslaught which ultimately failed to produce a winner.