A pulsating match in front of a capacity attendance had reached the fourth minute of injury-time when Scott McTominay used his hip to deflect a knocked-on corner from John McGinn into the net.
An old-fashioned Hampden roar greeted the announcement that six minutes were to be added on at the end. Clarke felt the noise "spooked" Israel.
McTominay's late strike was the first time in the evening that Scotland had been ahead after Israel twice took the lead, the second time moments after the hosts had equalised for the first time through McGinn’s goal on the half-hour mark.
The win keeps the Scots firmly on course to secure a World Cup finals play-off place after a timely run of winning form. Two more victories, against the Faroe Isles on Tuesday and Moldova next month, will secure second spot in Group F with one game to spare against Denmark.
Lyndon Dykes also saw a penalty saved by former Hibs goalkeeper Ofir Marciano on the stroke of half-time but redeemed himself with Scotland’s second equaliser ten minutes after the interval.
Referee Szymon Marciniak consulted the pitch-side TV monitor to judge whether he had been guilty of a high foot but allowed the goal to stand having originally booked the striker for dangerous play. The yellow card will now be rescinded.
“I don’t know if we’ll ever see that again,” said Clarke. “It was always going to be crucial and VAR’s been kind to us.
“When he disallows the goal I become a little bit nervous. The guys in the technical area are telling me it was clean contact. I’m praying that the referee goes and has a look – which he does.
“Listen, it’s a clean goal, and fantastic for Dykesy to get it after missing the penalty. His character showed, I think, in the second half, how good he is for us.
“Good for him to get the goal and we just kept plugging away. And then at 90 minutes, six minutes goes up and the crowd give that roar. I don’t know if it spooked the Israelis.
“But it certainly seemed to give us an extra bit of impetus and we get the winner and it turns out to be a great night. You see what it means for the supporters at the end of the night which was great for us.
“It’s important for the players to feel that love coming off the terraces. I keep saying that this group of players, every time we get together, every time we get to the pitch, they want to be successful for their country – and they showed that in the second half.”
Clarke was unusually emotional at the end and even started a Jose Mourinho-esque run down the touchline before thinking better of it. He was once assistant manager to Mourinho at Chelsea.
“I managed to run so far and then I thought ‘where the f*** are you going!?’” said Clarke. “Then I just wandered back into the technical area. He (Mourinho) would’ve been a way further…
“It was a good time to score a goal and then see the game out. In the context of the group, it's a big three points,” he added.
“It puts us in control of the position. But you’ve got to put the lid on it because it’s only three points and we need three more on Tuesday against the Faroes. That’s the next target, another cup final.”