The 23-year-old Hoffenheim attacker, who made his debut for the national side in Turkey last week, came off the bench to smash home an 83rd-minute winner in Vienna.
Austria gave a good account of themselves and looked set to earn a memorable draw when Aleksandar Dragovic found the net from the penalty spot after David Luiz had given the visitors the lead in the 64th minute.
However, Firmino's late strike earned a hard-fought victory for the South Americans, who are on a six-match winning run since Dunga re-assumed the reins from Luiz Felipe Scolari following a disappointing home World Cup, while Austria fell at the final hurdle in their bid to go through 2014 unbeaten.
Thiago Silva will not have been placated after admitting his frustrations at being replaced by Neymar as captain of the national team, with Dunga choosing to name the same XI that thumped Turkey 4-0 last week and leaving the deposed skipper among the substitutes once again.
Austria started brightest, though, and Rubin Okotie had the ball in the back of the net after six minutes.
It would have been the ideal start for the hosts had the striker not brazenly used both hands to slap the ball beyond goalkeeper Diego Alves from a Zlatko Junuzovic free-kick.
Despite Brazil enjoying the lion's share of possession, it was Austria who continued to pose the biggest threat in front of goal in the early stages – Filipe Luis hurling himself in the path of a dangerous-looking Florian Klein strike.
Okotie came close to opening the scoring once again in the 26th minute when he headed against the left upright, although the flag had already been raised for offside.
Miranda pulled up with what appeared to be a hamstring problem soon after, affording Silva a chance to prove his worth to coach Dunga.
Brazil's first shot on target did not materialise until after the half-hour mark, when Oscar saw a speculative low effort comfortably collected by home goalkeeper Robert Almer, before Luiz sent a 25-yard free-kick whistling over the crossbar.
The visitors ended the half in the ascendancy, but their appeals for a penalty when Danilo was barged over in the box went unrewarded.
Brazil continued to see a greater share of the ball after the interval, but struggled to carve out any clear-cut opportunities.
However, Luiz finally broke the deadlock in the 64th minute when he got the better of marker Stefan Ilsanker and evaded his marker to head home from a corner.
Dragovic levelled matters from the spot 11 minutes later after Andreas Weimann was felled by Oscar in the box, the equaliser representing the first goal Brazil have conceded in Dunga's second spell in charge.
Yet Firmino had the final say - firing in a swerving 20-yard effort seven minutes from time to announce his arrival on the international stage.
The reportedly disgruntled Silva finally got his way in injury time, when Neymar handed over the captain's armband after being substituted.
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