The Azzurri were already assured of top spot in Group J and were in control at the break in Zenica thanks to Francesco Acerbi's first Italy goal and Lorenzo Insigne's effort.
Andrea Belotti's fine strike early in the second half ensured Italy retained their 100 per cent record, while Bosnia-Herzegovina already knew they were destined for the play-offs.
It means Roberto Mancini breaks new ground as the first Italy coach to oversee 10 straight wins, while it also marks the first time they have won six away games in a row.
Gianluigi Donnarumma was sharp to keep out Ermin Bicakcic's close-range header and defender Zoran Kvrzic made a great block to deny Insigne before Italy went in front.
The visitors worked the ball into the box and Nicolo Barella teed up Acerbi, who showed quick feet before rifling into the bottom-right corner.
Things got better when Belotti wormed his way into the area to pick out Insigne and his effort found the bottom-left corner despite not making clean contact.
Donnarumma made fine saves from Gojko Cimirot and Adnan Kovacevic to protect the two-goal lead and Belotti put the match to bed after 52 minutes.
The Torino forward raced onto Barella's throughball and his sweetly hit 20-yard effort curved past Ibrahim Sehic.
Kovacevic was perhaps fortunate to escape with just a yellow card for a challenge that caught Belotti high and Mancini made the rare decision to sub a keeper when Pierluigi Gollini made his Italy bow late on, as the closing stages passed largely without incident.
What does it mean? Mancini has history-making Italy in great shape
After the ignominy of missing out on the 2018 World Cup under previous boss Gian Piero Ventura, Mancini has quietly gone about rebuilding Italy's shattered confidence. In Federico Bernardeschi and Insigne they have creativity in forward positions, while Jorginho, Sandro Tonali and Barella are a work in progress in midfield. Tougher tests lay in wait but a winning mentality has returned to the Azzurri.
Barella the fella making Italy tick
There were a lot of things to like for Italy, but Barella was particularly impressive pulling the strings in the heart of midfield. The Inter star finished the evening with four key passes and two assists.
Dzeko has his Ed-in his hands
Having spent the past four years in Serie A, Edin Dzeko is no stranger to Italy, but he unusually snatched at his chances and was unable to hurt the visitors.
Italy will look to make it 10 wins from 10 in Group J when they host Armenia in their final qualifier on Monday, while Bosnia-Herzegovina's last group game is at Liechtenstein on the same day.