Orient then had Skipper John Mackie sent off on 59 minutes before the home side looked set to cause a shock when Gary Hooper struck three minutes from the end - but livewire Sharp popped up with his last-gasp leveller.
Scunthorpe were quick out of the blocks and Jermaine Beckford shot wide from 16 yards but gradually Orient started to take a vice-like grip in midfield.
With their frontline pairing of Hooper and Ryan Jarvis causing problems the home side should have had a penalty when Steve Foster appeared to clip Hooper as he was about to shoot.
As Orient continued to press Hooper and Craig Easton tested visiting keeper Joe Murphy and, as the pressure mounted, Luke Gutteridge, Hooper and Corden all went close to breaking the deadlock.
So it was no surprise when Orient deservedly went ahead on 41 minutes. Justin Miller set Shane Tudor down the right with an intelligent pass and the winger's cross fell to Corden who slammed it home.
Scunthorpe responded early in the second half when MacKenzie swept home a leveller from eight yards after Beckford had surged down the left.
Orient looked to have suffered a hammer blow when Mackie saw red for hauling down Beckford on 59 minutes.
Scunthorpe were denied by the brilliance of Glyn Garner in the home goal as he kept out efforts from Andrew Crosby and MacKenzie before Hooper looked to have won the game when he hooked the ball home on 87.
But deep into stoppage time, Orient failed to clear a corner and Sharp crashed home a leveller.