Jeff Hendrick's deflected effort gave the Republic the perfect start as they held out until half-time despite almost constant Serbian pressure.
Filip Kostic levelled on 62 minutes and Dusan Tadic scored a controversial penalty seven minutes later.
The Irish looked beaten but Murphy headed the equaliser on 81 minutes.
Murphy's goal ended his long drought for the Republic as the Newcastle striker, 33, finally broke his international duck in his 24th appearance.
The Republic failed to build on the ideal start of Hendrick's third-minute goal as his low shot deflected in off Chelsea defender Branislav Ivanovic. It was a first international goal for Burnley's recent record signing from Derby.
For the remainder of the contest, the Republic struggled to retain possession as Kostic and Southampton striker Tadic caused the visitors no end of problems.
Only poor Serbian finishing and a splendid save by West Ham keeper Darren Randolph from Kostic kept the Irish in front by the interval.
Serbia's deserved equaliser came on 62 minutes as Kostic slotted from close range after Tadic had outfoxed John O'Shea in the Republic penalty area.
With the home side continuing to run at the ragged Irish, they were awarded a highly-debatable penalty with Jonathan Walters' contact on a charging Kostic just inside the area looking minimal.
Tadic dispatched the penalty into the roof of the net and the Republic were fortunate not to go 3-1 in arrears three minutes later as substitute Andrija Pavlovic hammered against the crossbar after Randolph spilled a Filip Mladenovic shot.
But the Republic summoned up a response, with Walters having an effort ruled out for offside and James McClean going close before Murphy powered in a headed equaliser as he connected with a Brady corner.
Given the Republic's defensive nervousness all night, it was no surprise that they had further late scares with Seamus Coleman desperately having to clear off the line after Randolph had blocked an Ivanovic effort.
The West Ham keeper rode his luck at times but made a couple of vital saves during stages of the contest when the Serbian were threatening to over-run the visitors.
In truth, few of his outfield colleagues were particularly impressive in a display where the Republic were guilty of squandering possession and resorting to long balls on too many occasions.
After the first round of fixtures, the Republic sit behind leaders Wales and Austria, who won against Moldova and Georgia respectively.
Martin O'Neill's side return to World Cup qualifier action next month, hosting the Georgians on Thursday, 6 October before an away date in Moldova three days later.
The busy autumn qualifying schedule concludes away in Austria on 12 November with Ireland's Group D action resuming against Wales in Dublin on 24 March.
Match ends, Serbia 2, Republic of Ireland 2.
Second Half ends, Serbia 2, Republic of Ireland 2.
Foul by Dusko Tosic (Serbia).
Jonathan Walters (Republic of Ireland) wins a free kick in the defensive half.
Substitution, Republic of Ireland. Ciaran Clark replaces Shane Long because of an injury.
Delay over. They are ready to continue.
Delay in match Shane Long (Republic of Ireland) because of an injury.
Foul by Aleksandar Katai (Serbia).
Jonathan Walters (Republic of Ireland) wins a free kick in the defensive half.
Glenn Whelan (Republic of Ireland) is shown the yellow card for a bad foul.
Dusan Tadic (Serbia) wins a free kick in the attacking half.
Foul by Glenn Whelan (Republic of Ireland).
Corner,  Serbia. Conceded by Daryl Murphy.
Attempt saved. Branislav Ivanovic (Serbia) right footed shot from the centre of the box is saved in the centre of the goal.
Attempt missed. Dusan Tadic (Serbia) right footed shot from outside the box is close, but misses to the left. Assisted by Dusko Tosic following a corner.
Attempt blocked. Dusko Tosic (Serbia) right footed shot from outside the box is blocked. Assisted by Nemanja Gudelj.
Corner,  Serbia. Conceded by Richard Keogh.
Aleksandar Katai (Serbia) is shown the yellow card for a bad foul.
Foul by Aleksandar Katai (Serbia).
Jonathan Walters (Republic of Ireland) wins a free kick on the right wing.
Foul by Dusko Tosic (Serbia).
Shane Long (Republic of Ireland) wins a free kick on the right wing.
Substitution, Serbia. Aleksandar Katai replaces Filip Kostic.
Goal!  Serbia 2, Republic of Ireland 2. Daryl Murphy (Republic of Ireland) header from the centre of the box to the bottom left corner. Assisted by Robbie Brady with a cross following a corner.
Corner,  Republic of Ireland. Conceded by Antonio Rukavina.
Corner,  Republic of Ireland. Conceded by Predrag Rajkovic.
Attempt saved. Shane Long (Republic of Ireland) right footed shot from the centre of the box is saved in the centre of the goal. Assisted by Daryl Murphy with a headed pass.
Attempt missed. James McClean (Republic of Ireland) header from the centre of the box is too high. Assisted by Robbie Brady with a cross.
Attempt blocked. James McClean (Republic of Ireland) header from the centre of the box is blocked. Assisted by Seamus Coleman with a cross.
Substitution, Serbia. Dusko Tosic replaces Filip Mladenovic.
Substitution, Republic of Ireland. Daryl Murphy replaces Jeff Hendrick.
Corner,  Republic of Ireland. Conceded by Matija Nastasic.
Offside, Republic of Ireland. Jeff Hendrick tries a through ball, but Jonathan Walters is caught offside.
Matija Nastasic (Serbia) is shown the yellow card for a bad foul.
Foul by Matija Nastasic (Serbia).
Shane Long (Republic of Ireland) wins a free kick on the left wing.
Andrija Pavlovic (Serbia) hits the bar with a left footed shot from the centre of the box.
Attempt saved. Filip Mladenovic (Serbia) right footed shot from outside the box is saved in the centre of the goal. Assisted by Dusan Tadic.
Substitution, Republic of Ireland. Stephen Quinn replaces Stephen Ward.
Goal!  Serbia 2, Republic of Ireland 1. Dusan Tadic (Serbia) converts the penalty with a left footed shot to the centre of the goal.
Daryl Murphy's first international goal earned the Republic of Ireland a lucky draw in their opening World Cup qualifier in a rain-lashed Belgrade.