The unbeaten Swiss moved top of Group I, with 16 points from eight matches, but only ahead of Romania on goal difference.
Israel are third on 12 points with all three countries having two games left.
Vargas came within inches of doubling his tally five minutes after his goal when he thundered a shot off the crossbar.
As a result, Israel play four games in the space of nine days to complete their qualifying campaign.
The Swiss face Kosovo in Basel on Saturday and then complete their campaign away in Romania on Tuesday.
Persons:
Israel, Shon Weissman's, Ruben Vargas, Edimilson Fernandes, Weisman’s, Fernandes, Noah Okafor, Vargas, Sean Goldberg, Dor Turgeman’s, Turgeman, Yann Sommer, Mark Gleeson, Toby Davis
Organizations:
European Championship, Kosovo, Swiss, Palestinian, Hamas, Romania, Puskas Academy Arena, Thomson
Locations:
FELCSUT, Hungary, Switzerland, Germany, Swiss, Romania, Israel, Kosovo, Budapest, Andorra, Basel, Cape Town