The hosts had huffed and puffed without reward until the 71st minute when central defender Baudry met a corner from Dean Cox and sent an unstoppable header into the top right-hand corner from eight yards.
However, in front of the O’s biggest attendance of the season, Harris scored just six minutes later to win the Blades the point they needed to guarantee a play-off place.
He ran onto a clearance and powered a shot goalwards that saw the ball take a wicked deflection past Orient goalkeeper Alex Cisak and into the net.
Cisack had earlier kept his side in the game in the opening period, pulling off a series of outstanding saves to deny Jason Holt and Jamie Murphy from point-blank range, while also extending to full stretch to palm away an effort from Che Adams.
Orient still have a chance of avoiding the drop but have to win at Swindon next week and hope that other results go in their favour.