Dragan Elkaz was the hero for Australia, scoring 17 of his 25 points in the fourth quarter while Tom Fullarton (ten points) and Samson Froling (four points, ten rebounds, two blocks) were also impressive down the stretch.
Fullarton kick-started the run with a three-pointer at the beginning of the fourth period before Elkaz drained three three's in quick succession to give Australia the lead with six minutes remaining.
Dzanan Musa was able to answer for Bosnia and Herzegovina, seemingly never tiring as he continually drove to the rim, collecting 12 of his game-high 34 points from the free-throw line.
However, Fullarton did not back down in his defence on Musa and costly missed shots from the charity stripe late kept Australia in front.
Froling then hit an important runner off the glass to extend the buffer to three points before Fullarton hit a lay-up on Australia's final offensive possession.
In the last 30 seconds, a mad scramble ensued as the Europeans missed an incredible seven shots, including a buzzer-beating three-point attempt which would have sent the game into overtime.
The victory will see Australia face Spain in the Quarter-Finals on Thursday night with Spain currently undefeated in the tournament.
From the outset of the Round of 16 clash, the Aussies looked switched on as they drilled their first five field goals from two-point range while starting the game with a 6-0 run.
They maintained that hot shooting through the opening period, hitting 9-15 overall to hold that six point advantage to quarter-time.
Bosnia and Herzegovina found their running game shortly after the first break though and as their interior defence tightened, Australia struggled to find a way to penetrate.
Their offence dried up, scoring only six points for the term to find themselves trailing at half-time.
Elkaz was able to give Australia an offensive injection coming out of the main break before Kyle Zunic (three points, two assists) ran several pick and roll's to perfection with big man Mate Colina (six points).
Musa then looked set to break the game open for Bosnia and Herzegovina as he got a couple of his opponents such as Matthew Johns (two points, five rebounds, four blocks) in foul trouble but when Elkaz matched his offensive output at the other end of the floor, Australia had regained the ascendancy.
Click here to find the full box score while a replay of the game (except for the first term due to a FIBA technical issue) can be seen below.