Meanwhile, Stockport continue to look for their first home win of the season - they have drawn all six home League Two games this term.
While the O's made just the one enforced change, Stockport, who went into the game right at the bottom of League Two, saw four changes, with most notably club skipper Rob Clare and team captain Keith Briggs both being dropped.
Northern Ireland international Danny Griffin also came into the Stockport side for his first start of a so-far worrying season for the Hatters, he has just got over a long-term knee injury.
In driving rain, it was Martin Ling's side who created the first opportunity when Jabo Ibehre saw a near-post effort well saved at the second attempt by County goalkeeper Carl Ikeme.
Minutes later Stockport should have opened the scoring though, O's keeper Glyn Garner spilled a Matt Hamshaw cross, but the recalled Tes Bramble scooped a poor shot over the bar from ten yards.
Stockport did make the breakthrough on 22 minutes, however. New loan capture Gavin Strachan floated over a corner-kick, Michael Raynes headed the ball back into the danger zone and there was experienced centre-back Tony Vaughan who fired home a crisp angled half-volley from just inside the box.
Craig Easton tried an immediate response but his low drive fizzed just wide, while at the other end Bramble glanced a header just over the crossbar following good work down the right from Hamshaw.
Just a minute after the restart Bramble again went close when his deflected shot was well parried in the increasingly wet conditions by Garner.
Michael Malcolm was the next to tee up the fired-up Bramble, but once again Garner got the better of the giant striker with a fine point-blank save.
The visitors were on top going into the closing stages though, and on 71 minutes they got themselves back on terms when Alexander somewhat fortuitously scooped home following a smart cross from Shane Tudor.
The visitors later went closest to snatching all three points when Alexander hit the crossbar with a neat glancing header, but a draw was just about a fair outcome.