Napoli star Insigne volleyed in a brilliant equaliser soon after the interval and provided the pass for Marco Verratti's decisive 86th-minute strike as Roberto Mancini's men extended their winning run to five matches.
Edin Dzeko had given the impressive visitors a 32nd-minute lead – the first goal conceded by the Azzurri since October – and their performance merited at least a point.
It was not to be, though, as Robert Prosinecki's side lost ground on the top two in Group J, in which Italy have taken maximum points from their opening four matches.
Bullish from the beginning, Bosnia-Herzegovina might have been ahead as early as the 18th minute had Leonardo Bonucci not dived to brilliantly block Edin Visca's shot.
Dzeko provided the pass for that chance and roles were reversed for the opener, the captain turning home after winger Visca broke to the byline.
Italy restored parity in stunning style just over three minutes after the restart as Insigne lashed a first-time volley into the bottom-left corner from Federico Bernardeschi's corner.
Bosnia-Herzegovina bounced straight back onto the front foot with Giorgio Chiellini nodding an Eldar Civic header off the line, before Salvatore Sirigu thwarted Visca.
Chiellini's aerial prowess almost paid dividends at the other end in the 72nd minute, but Ibrahim Sehic positioned himself well to parry.
Italy funnelled their attack through one man, however, and Insigne unsurprisingly had a say in the winner, finding midfielder Verratti free in the box for a measured, curling finish into the bottom-right corner.
What does it mean? Azzurri strengthen grip on Group J
This was not a vintage performance from a team that has evolved encouragingly in recent months and their 3-0 weekend win in Athens was made to look less glamorous by Armenia's defeat of Greece elsewhere on Tuesday.
Regardless, it is a result that keeps Italy three points ahead of second-placed Finland and six clear of the teams outside the automatic Euro 2020 qualifying positions.
Insigne the leader of Azzurri attack
Italy might never have really clicked at Allianz Stadium, but that was not a necessity with Insigne scoring and assisting goals.
The 28-year-old is simply the most important figure in a forward line that is yet to really adopt a consistent look under Mancini.
Quandary over Quagliarella
It does not reflect well on Fabio Quagliarella that Italy scored so soon after his half-time withdrawal, nor that the front three appeared more fluid in his absence.
He tallied just nine touches and one shot of note in his 45 minutes on the pitch - head coach Mancini might have patience with peripheral displays from a young player, but not one of Quagliarella's 36 years.
Italy visit Armenia and Finland when qualifying resumes in September. Bosnia-Herzegovina head to Armenia following a home date with Liechtenstein.