Monday's (Tuesday AEDT) result, celebrated ecstatically in the cauldron of Red Star's Rajko Mitic stadium, left the Balkan nation top of Group D with 21 points from 10 games, two more than second-placed Ireland who booked a play-off spot with a 1-0 win at Wales.
It is the second time Serbia have reached the finals as an independent nation, having been knocked in the group stages of the 2010 tournament in South Africa and then missing the next three major international soccer championships.
Coach Slavoljub Muslin, a journeyman who had been in charge of a myriad of European clubs before he took over at Serbia for the qualifying campaign, was elated after the success.
"I expected a tough game against Georgia and suspected that the final step would be the most difficult," he told a news conference after the organisers set off spectacular fireworks which rocketed into the night sky over the Serbian capital.
"Georgia were the better side in the first half and I told the players to move up a gear after the break, though we had to keep it tight at the back too."
Prijovic grabbed the winner with a fine finish to delight raucous home fans, who lit several flares to celebrate.
It came after anything but the expected plain sailing for the home side, who looked disjointed for much of the contest after missing their first chance to qualify when they lost 3-2 at Austria on Friday (Saturday AEDT).
Serbia piled on the pressure from the offset but Georgia looked more likely to score in the first half and came close in the 17th minute when midfielder Valerian Gvilia forced a good save from Serbia keeper Vladimir Stojkovic.
The Serbians carved out their first chance in the 39th minute, as visiting keeper Giorgi Makaridze blocked Dusan Tadic's shot from a tight angle.
Jeered off by their supporters at halftime, Serbia came out with renewed purpose after the break and defensive midfielder Nemanja Gudelj fired narrowly over the bar in the 48th minute.
Shortly afterwards, striker Aleksandar Mitrovic saw his close-range header superbly saved by Makaridze and then sliced another wide of the far post as the home side threw men forward.
Ivanovic headed inches wide after a corner the 67th minute and Nemanja Matic had a shot cleared off the line moments later as Serbia besieged the Georgian goal with a flurry of crosses from either flank.
Prijovic finally broke the deadlock with a composed close-range finish after strike partner Mitrovic found him with a fine low cross following Ivanovic's darting run through the middle.
"Few fans believed after our barren eight-year run that we would qualify and all credit goes to the players, who left it all out there in every match," Muslin added.
"The fans played a big part too and I thank them."