After both teams hit the woodwork early in the second half, Edin Dzeko put the visitors in front when he fired into the roof of the net with a shot on the turn from a corner in the 57th minute in their League A Group One match.
Italy, playing their first match since beating Armenia 9-1 last November, won all 10 of their qualifying matches for Euro 2020 as well as a friendly against the United States for the longest winning run in their history.
The gap of 291 days between fixtures was Italy's longest since 1945-46.
Italy had dominated the first half against defensive opponents but failed to make much headway.
Armin Hodzic hit the post from a narrow angle for Bosnia after the break and Insigne almost immediately headed against the woodwork for Italy.
"We're disappointed, we tried until the end to score," said Mancini.
"There was a bit of tiredness but the boys did well. We just lacked a bit of inspiration,
"The important thing is that the team played well, the draw doesn't change anything in the judgement of the performance."