James Forrest scored a magnificent hat-trick as the Scots recovered from falling behind to a Beram Kayal strike for Israel.
McLeish’s side had to endure a nerve-shredding ending to the contest as Eran Zahavi cut the arrears for the visitors in the closing stages before Allan McGregor’s stunning stoppage time save denied them an equaliser.
The 3-2 win saw Scotland top Group C1, earning a place among the third pot of seeds in the draw for the main qualifying path for the Euro 2020 which takes place in Dublin on 2 December.
If they do not qualifying through that route, then as the highest ranked of the four Nations League C group winners, they will have a home tie in the one-off semi-final of the play-offs in March 2020, currently against Finland.
Norway and Serbia were the other League C group winners.
“Tonight I was quite calm,” insisted McLeish, despite the rollercoaster nature of the match. “I felt really strong about the team going into this game after the way they played to a high intensity in beating Albania on Saturday.
“We played it extremely professionally and to keep that high intensity tonight was a wee bit of a task. You saw some of them flagging because Israel had a couple more days rest than us.
“I’m so proud of the guys and we answered a lot of questions. It feels perfect to me because the goal was to win the game and we did that. We’ll look at the video and look to improve but the lads played to a fantastic intensity.”
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McLeish hailed the contribution of Celtic winger Forrest, who followed up his first two goals for his country in the 4-0 win in Albania with last night’s double, and Rangers goalkeeper McGregor.
“James’ goals over the two games are amazing,” said McLeish. “I complimented him after Saturday and tonight his finishing was unerring. To see them nestling just inside the post was fantastic.
“Any striker would be proud of that. He is in phenomenal form. Greegsy’s save was a fantastic, world class save at that time in the game. He’s done that all season.”
McLeish had come under fierce scrutiny during his second spell in charge of Scotland and savoured finishing 2018 on a high note.
“I’ve got to put it in perspective,” he said. “I was down after we lost the game to Israel in Haifa last month. But I’m always very conscientious because I want to do better.
“Criticism is fine, because it wants to make you do better. Sometimes it can be over the top. But we bounced back and showed resilience.
“There is more to come from this team. Now they have evidence of what they have done over the last two games.
“It’s given us a massive feeling of power and self satisfaction. We were written off and we’ve come through it against adversity. That’s a great part of your character when you can do that.
You don’t get a chance to rest on your laurels. You have to try to keep improving.
“It’s great to get the light at the end of the tunnel in terms of the play-offs for Euro 2020 and the catalyst for it was the performance in Albania.
“The players should be looking to be stronger mentally after those kind of performances.”