The Panthers sealed the win when defensive back Bricen Garner intercepted Youngstown State quarterback Hunter Wells' in the end zone on the Penguins' first possession in overtime.
Kicker Alex Kessman, who earlier in the fourth quarter missed a 28-yard field goal attempt, missed another 28-yard attempt as time expired that would have given Pitt the win.
Browne atoned for a costly fourth-quarter fumble that allowed the Penguins to tie the game by completing both pass attempts in overtime. In his first game for Pitt, Browne was an economical 17 of 24 for 140 yards and the lone touchdown.
Wells, who started nine games last season and led the Penguins into the FCS championship game, finished 18-for-32 for 311 yards, two touchdowns and the one interception.
The first half belonged to the Panthers. The final 30 minutes was all Penguins.
Running back Qadree Ollison capped an 11-play, 71-yard drive with a six-yard scamper to put Pitt up 7-0 in the first quarter.
On the Panthers' next possession, Ollison scored from five yards out to give the Panthers a 14-0 lead.
The Penguins had a chance to get on the board on their next possession, but Zak Kennedy missed a 43-yard field-goal attempt.
Pitt made the Penguins pay and drove the ball 74 yards in 16 plays. A.J. Davis punched the ball in from the 1; after eating up 8:47, the Panthers were up comfortably, 21-0.
But Youngstown State and third-year coach Bo Pelini, who were coming off a 12-4 season and a loss in the FCS Championship game, had plenty of fight left.
On their second possession of the third quarter, the Penguins drove 90 yards with Tevin McCaster scoring from a yard out to make it 21-7.
Kessman's first missed field goal attempt kept the score at 21-7 and set the stage for the Penguins to get within one score.
Wells connected with Christian Turner on a 25-yard scoring strike to complete an 80-yard drive, and on the Panthers' next possession, Browne fumbled at the Pitt 42-yard line after a sack.
Two plays later, Wells and Turner struck again, this time for a 42-yard score to tie the game at 21.
With his team reeling and being knocked back on its feet, Pitt coach Pat Narduzzi didn't want to give the Penguins a last shot at winning with less than two minutes to go.
With fourth-and-1 on the Pitt 46-yard line, Narduzzi rolled the dice, and Davis gained two yards to keep the drive alive.
But Kessman's field goal attempt was no good and the game went into overtime.