Lauren Steuer played a perfect corner to Brady who was back post and headed the ball past the feet of Purdue goalkeeper Jenny Bradfisch.
"It is a big win for everybody on our team because nobody here has been able to beat Purdue in their career," said Brady as the Buckeyes snapped a seven-game losing streak against the Boilermakers.
With the win, the Buckeyes (13-2-2, 6-1-1 Big Ten) remain atop the Big Ten with 19 points. It also marked the second consecutive home game Ohio State has won in the final minute after Lauren Granberg scored the game-winner against Northwestern with 48 seconds remaining two weeks ago.
The Buckeyes had their first opportunity of the half in the 37th minute when Ashley Bowyer played a nice ball to Steuer who had made a run down the left side but her shot from 12 yards out drifted wide left. Less than five minutes later, Bowyer almost found the back of the net but Bradfisch was able to make the diving save on the header.
The next Ohio State opportunity came in the 41st minute when a Steuer cross found Karin Sendel in the center of the box but the shot by the freshman sailed just high as the teams would head into the break in a scoreless tie.
For the half, the Buckeyes tallied five shots compared to two for the Boilermakers and held a 3-0 advantage in corners.
The Buckeyes continued to earn chances starting the second half but Bowyer's free kick from the top of the Box sailed just high.
As the teams continued with their physical play Purdue recorded their best opportunity of the game when Vanessa Ibewuike crossed the ball from the right wing through the box but Cassie Dickerson was able to clear the ball out before a Purdue attacker reached it far post.
Neither team would get many opportunities from there on out as the Buckeyes headed into their fourth overtime game of the season.
As the Buckeyes attacked less than a minute into overtime, Steuer's header was deflected high to set up the game-winning corner.
"Purdue is a tremendous team and they have been a great battle for us over the years," said head coach Lori Walker. "One of our mottos today was, 'it ends today.' This team has used our 2004 team as a model. For us to finally steal a win back from Purdue is huge on a day like today at home."
For the game, the Buckeyes outshot Purdue 14-3 and held an 8-1 advantage in corners.
Buckeye goalkeeper Lauren Robertson also earned her second shutout of the weekend and 12th of the season against the Boilermakers.
Ohio State will have a week off before taking on Indiana on Senior Day next Sunday at Jesse Owens Memorial Stadium. First kick is set for 1 p.m.