Ben Doak missed a penalty with two minutes left which would have secured away double-header victories and helped the young Scots take a giant step to the Euro 2025 Finals.
Scot Gemmill's side had stunned Belgium in Flanders last Friday with a 2-0 win, but had to rely on a brilliant performance from Ipswich keeper Cieran Slicker in Budapest last night who made a string of first-half saves.
Scotland improved after the break and were handed the chance to win it when sub Dire Mebude was brought down in the box.
But the Liverpool starlet saw his spot kick saved and Scotland had to settle for a point. They remain in second place in the group though at the halfway stage in the battle to reach the Finals in Slovakia. And Gemmill tipped Doak to quickly recover.
He said: "Ben will be fine. He's young and he will bounce back fast. It's a good learning experience for all of them and they are all disappointed because we came so close to getting another victory and I have just reminded them of that.
"We have to think of the campaign from the Spain game to this game and the fact we have gone away to Belgium and Hungary and not conceded a goal.
"There were a lot of really good things and they have shown a lot of character and we are sitting with ten points. "We can be really optimistic at the next set of games. I know it's tight in the group. It's exciting and great for the players and I am really proud of the performances from them so far.
"Right now it's easy to underappreciate how good the two games have been because both Belgium and Hungary are really good teams as you could see in the first half.
"So it was really good to see the character and attitude in the team to be able to withstand that and do better in the second half and obviously have that chance to win it.
"It's one of those isn't it, we came so close to winning it, so right now it's not sitting good but it's important to keep everything in context.
"There were things that needed to be corrected at half time and the players did that and we were much better in the second half."
Gemmill opted to throw Rangers' Robbie Fraser straight into the team after only being called up after the Belgium game and he was the only change from that win in Roeselare when he replaced the injured Matthew Anderson.
Hungary, managed by ex-Fulham and West Brom midfielder Zolten Gera, dominated the first half and Scotland had Slicker to thank for keeping them in it.
He produced three great saves in a row with two of them from Dunajska Streda midfielder Milan Vitalis.
Then from the resulting corner, Slicker spread himself to prevent a volley from Mihaly Kata, who plays for host club MTK Budapest, from finding the target with the young Scots suddenly on the ropes.
And Slicker made another fine block from Vitalis from close range as the home side turned the screw and Scotland sagged.
Just before the break Slicker again dived full length to stop Lommel striker Zalan Vancsa from opening the scoring.
Yet Doak nearly gave Scotland the lead when he embarked on one of his trademark explosive runs when he ran 60 yards before he got his shot away which was kept out by keeper Krisztian Hegyi, who plays for Stevenage.
Scotland improved after the break and stemmed the flow of play towards their goal and Hungary struggling to assert the same levels of dominance they did before the break.
And a rare Scotland attack saw Doak create a chance for sub Mebude, but the Westerlo striker didn't catch his effort right.
But Mebude was then desperately unlucky when Doak teed him up after some trickery from a short corner and the former Manchester City and Rangers striker's effort was deflected into the side netting.
Then he was brought down in the box as he darted into the box by Hungary captain Peter Barath to hand Scotland that huge chance to claim another victory.
However, Doak was denied by a brilliant save by Hegyi who dived full length to his right to keep the spot kick out.
