Miles Hippolyte put the Hatters in front after just five minutes, taking Paddy Madden's flick and firing home from an angle.
Beckles then found a winner after Hinchcliffe tipped Idris El-Mizouni's piledriver onto the crossbar, but the forward was on hand to tuck home the rebound.
Orient extend their lead at the summit to five points over second-place Stevenage while Stockport, who saw their seven-match unbeaten run come to an end, remain 14th.
Story of the match
After just a couple of minutes had passed, the Hatters had the first chance of the match as Madden sent a through ball to Hippolyte, whose low shot crashed off the legs of Lawrence Vigouroux.
The hosts had further chances through Madden and Wootton that Vigouroux kept out with his hands and legs, respectively. At the other end, Charlie Kelman saw his shot blocked.
On 20 minutes, Orient were level. A long throw from the left found Beckles and he flicked to Pratley, who headed past Hinchcliffe into the top corner.
Stockport nearly regained the lead as an inswinging corner by Chris Hussey was met by Akil Wright at the near post only to see Dan Happe clear off the line before Macauley Southam-Hales recycled the ball for Collar, who headed over.
The Hatters had the initial chance of the second half as a looping delivery from the right by Southam-Hales that Vigouroux caught on his line.
A counterattack by the O's saw Paul Smyth cut through on the left only to see his shot smothered by Hinchcliffe.
Seven minutes after the restart, Theo Archibald saw his strike blocked and George Moncur found Smyth on the right, his tight-angle shot saved by Hinchcliffe.
On the hour mark, the O's made a double change as Moncur and Smyth made way for Shad Ogie and Aaron Drinan.
Drinan did well to find Kelman, the Stockport defense needing to clear, which resulted in a corner.
That corner, Orient's first of the match, put them in front. Archibald sent it in and it was only cleared as far as El Mizouni, who thundered a shot that Hinchcliffe did brilliantly to keep out only to see Beckles put home the rebound.
With 12 minutes remaining, Hinchcliffe was called on again, stretching to parry a low shot by Kelman around the right post.
Ogie did well to dispossess Madden with ten minutes remaining and cleared into the box on 87 minutes before Kelman was replaced by Ruel Sotiriou.
During six minutes of stoppage time, Vigouroux denied Wootton's header to ensure all three points would head back to East London.
Player of the match: Omar Beckles
Scored the winner and looked dangerous throughout, having additional chances to add to his and Leyton Orient's tally.