Szoboszlai's strike ricocheted in off the post in the 80th minute, leaving Turkey goalkeeper Ugurcan Cakir with little chance, after the home side had raised their game in the second half following a poor start.
Playing without spectators in the central Turkish city of Sivas, Hungary created a series of goalscoring opportunities in the first half but failed to capitalise, with Roland Sallai and Filip Holender forcing early saves from Cakir.
Szoboszlai hit the post with a shot from the edge of the area and had another effort blocked.
Turkey began the second half much improved, with Burak Yilmaz having a great opportunity to put the hosts ahead in the 52nd minute when he was put through on goal, only for Hungary keeper Peter Gulasci to block his effort.
Turkey's Hakan Calhanoglu fired over the bar from the edge of the penalty area, but Hungary gradually reasserted themselves and hung on for victory after Szoboszlai's goal.
Aleksandar Mitrovic clawed a goal back for Serbia in the 78th minute, before Dzyuba sealed Russia's victory with a second goal.
Turkey will play Serbia and Russia face Hungary on Sunday.