Malaga went in front after 19 minutes as Isco's shot from outside the area took the slightest of deflections off Ion Ansotegi and left Bravo with no chance.
The hosts seemed set to move back into third place in the La Liga when Mikel Gonzalez was dismissed for a second bookable offence just before half-time, but Sociedad were level three minutes after the break as Xabi Prieto's shot from distance found the bottom-left corner of the net.
The home side then threw everything at the Sociedad goal but found Bravo in unbeatable form as he saved brilliantly from Jesus Gamez, turned Salomon Rondon's overhead kick onto the post and then again denied the Venezuelan with a stop from point-blank range three minutes from time.
Malaga were aiming for their seventh home win in eight games and after an early scare when Antoine Griezmann's shot was deflected behind, they dominated proceedings.
Isco has become accustomed to scoring spectacular goals this season and his opener was no different, although the slight touch it took off Ansotegi may have helped its trajectory as it looped into the far corner of Bravo's net.
Santi Cazorla nearly made it two from similar range but his effort was pushed away by Bravo and Sebastian Fernandez should have doubled Malaga's lead but he headed just wide from Nacho Monreal's cross two minutes before the break.
However, there was time for Sociedad to be reduced to 10 before half-time as Gonzalez's late sliding challenge on Fernandez brought him a stupid second yellow card.
Despite the man disadvantage though Prieto levelled matters as a free-kick on the edge of the area broke kindly back to him from Joris Mathijsen's clearance and he duly dispatched it low beyond third-choice Malaga keeper Ruben Martinez.
Fernandez came close with his head again minutes later before Bravo made a stunning save to deflect Gamez's volley away for a corner.
The chances kept coming for the hosts as Ruud van Nistelrooy's header was well held by Bravo, but at the other end Malaga were thankful to Ruben for keeping them level as he made a smart stop to deny Prieto his second.
Rondon then tried his luck with a bicycle kick that Bravo got enough on to turn it behind, but the Chilean's save from the same player moments later was his best of the day as he somehow pushed Rondon's powerful volley from inside the area over the bar.
And it was not to be Malaga's day as even when they did beat Bravo, the assistant referee's flag got in the way when Martin Demichelis was wrongly ruled offside when he poked the ball home in stoppage time.