Goals by Gil Vermouth in the 36th minute, Omer Damari (45)and Eran Zahavi (70) secured the win as Bosnia played most of the second half with 10 men after Toni Sunjic was sent off in the 48th minute for bringing down Damari.
Israel have never qualified for the European Championship but are now top on nine points from three games after Wales drew 0-0 in Belgium.
Their debut at Haifa's new Sammy Ofer Stadium could barely have been more positive, as coach Eli Guttman's men dominated from the start and struck at the right time to shut out the Bosnians, who were without injured Manchester City forward Edin Dzeko.
The Bosnians were second best for most of the match and when they looked like getting on level terms, Israel struck to dampen their spirits.
Vermouth, of Hapoel Tel Aviv, was perfectly set up by striker Tal Ben Haim for the first goal and Damari added the second just before halftime after being put through by Vermouth.
Once Sunjic was sent off, with Damari hurtling goalwards, the match was all but over and Maccabi Tel Aviv's Zahavi earned the plaudits of the ecstatic crowd.
"I am very happy but I am keeping my feet on the ground and am maintaining a modest outlook. The road to qualification is still a very long one," Guttman said.
The defeat is likely to pile more pressure on Bosnia's underfire coach Safet Susic, criticised for his tactics and team selection during an early World Cup exit as well as in the opening three Euro 2016 qualifiers. Bosnia have two points.
(Additional reporting by Zoran Milosavljevic; Editing by Mark Meadows)
