McLeish saw his players withstand a physical performance from their hosts to clinch a morale-boosting victory thanks to Matt Phillips’ winner three minutes into the second half.
The result provided McLeish with a boost following the 1-0 defeat by Costa Rica last week in the first match of his second spell in charge.
He was particularly pleased by the manner of last night’s win, which saw Scotland recover from the blow of seeing skipper Charlie Mulgrew’s penalty saved by goalkeeper Peter Gulacsi shortly before the interval.
“The evidence is there for them now,” said McLeish. “We can work on that. They have won a game and we can use that to empower them, the confidence they can take from that. Hungary have some good players, dangerous players, but our players handled them well.”
McLeish admitted the result was of great personal significance. No Scotland manager since Berti Vogts has lost their first two matches and McLeish had already come under fire from some quarters for picking an inexperienced side against Costa Rica.
“It is a big night for me,” he said. “I was asked earlier if the criticism was over the top. I must admit I didn’t see too much criticism. If I felt it was out of order I would say that.
“But we didn’t get our rhythm in the first half on Friday because so many players hadn’t played together before. Tonight there was more familiarity between players who have a big future.”
“It was great to finally put the ball in the back of the net,” added McLeish. “By and large, we controlled the game. We played to a very decent level.”
There was a strong performance on the right of a three-man defence by 22-year-old debutant Jack Hendry and Aberdeen’s Scott McKenna also coped well again alongside Mulgrew.
“It was good to introduce new caps,” said McLeish, with Wolves left-back Barry Douglas also making his first international appearance in the second half. “I thought our two young central defenders were really good. Jack was coming in for his first game and concentrated the whole game. Scott McKenna probably had an even better game tonight than he did on Friday.
“You have to control your emotions and the players did that. We missed a penalty and you wonder how they are going to react. Are the heads going to go down? They didn’t. When you get the goal are they going to switch off? They didn’t.”
Winger Phillips, playing instead through the middle, scored his first goal on his 11th international appearance.
“Matt is a versatile player,” said McLeish. “He would say that centre forward is not his absolute best position but we needed a goal or two. We got one and that was enough.”