Northern Ireland will host Slovakia next month for a place at the postponed tournament next summer after it too prevailed on penalties after a 1-1 draw against Bosnia and Herzegovina.
Scotland is now one game away from its first major tournament in 22 years. It next travels to Serbia on 13 November (AEST), but was pushed all the way after neither side could find the net in 120 minutes at Hampden.
"It its another step forward to hopefully making a bit of history for the country," Scotland boss Steve Clarke said. "Qualification is still in our hands. We know we've got a tough game but it is one that we'll look forward to.
"I can feel there is something building. It's another game unbeaten and you can feel the elation in the dressing room."
Clarke's preparations were thrown into chaos on Thursday (AEST) when three key players were ruled out.
Stuart Armstrong tested positive for coronavirus, while Kieran Tierney and Ryan Christie were forced into self-isolation for being in close contact with the Southampton midfielder.
The goal-threat and creativity offered by Armstrong and Christie was badly missed as Scotland laboured to breakdown an Israeli side that has won just one of its past 10 internationals, including a 1-1 draw when it visited Hampden in the UEFA Nations League last month.
Manchester United's Scott McTominay spurned the best chance of the game when he headed wide when completely unmarked from a corner just before half-time and Liam Cooper's header hit the post with the last action of extra-time.
However, McTominay redeemed himself in the shootout as all five of Scotland's penalty takers found the net, while David Marshall saved from Eran Zahavi.
"Saving the first penalty breaks them a bit and gave us confidence," Marshall said.
"The last four penalties from us were a real high standard – the lads have practiced every day. It's more relief than anything."