On loan from Bristol City, Bridges showed the skill that made him a Premiership force with Leeds United and Sunderland when meeting Chris Lumsdon's driven free-kick to steer the ball into the net for an equaliser with 11 minutes left.
With half the team aged 21 or under, Posh made most of the first half running against a side that had hit 16 goals in its previous five outings and , after 20 year-old Ryan Semple had a shot cleared from the line, teenager Jamie Day whipped over a corner that Mark Arber glanced just wide.
Ex-Norwich youngster Danny Crow was also a handful as Posh pumped over crosses from either flank and, even though the 19 year-old's fiercely struck effort took a deflection, Carlisle goalkeeper Kieron Westwood still had to move smartly to gather the ball safely.
Playing three up front, Carlisle were equally adventurous, with Chris Billy moving forward from midfield to force home goalkeeper Mark Tyler into a flying save to keep out a shot heading for the top corner of the net.
It was even better for Posh after the break with Jamie Hand's 55th minute run on the right producing a cross for Semple to strike. The effort was blocked but Crow was ideally positioned to pounce on the rebound and prod the ball into the net for his seventh goal of the season.
Carlisle boss Paul Simpson committed all three substitutes in quick succession in a bid to save the game but even Simon Hackney, sent on in place of Derek Holmes, found it hard to bring 15-goal Kevin Hawley into the game.
In the end it was the experience of Bridges that helped to salvage a point, although home manager Mark Wright will probably have a few harsh words for Hand, who put pressure on his side with a needless foul to give away the all-important free kick that was converted at the far post.