It looked bleak for Martin Ling's team when Wrexham's captain Darren Ferguson put the visitors ahead on 53 minutes.
The Welsh side defended doggedly and looked set for their first away success of the season - until Alexander's late strike.
Orient had dominated the first half but failed to test Wrexham's rooky goalkeeper Michael Jones.
In the first 10 minutes Craig Easton sent in a snapshot from distance that the teenage keeper held and Alexander failed to hit the target when he spun and sent a close range shot over the bar.
After that Orient huffed and puffed but Wrexham stood firm.
All they had to show for their pressure before the break was an effort from Daryl McMahon and a long-range shot from Matthew Lockwood on the stroke of half time.
After the interval it was Wrexham who made the all important breakthrough with a goal out of nothing.
Darren Ferguson so impressive all afternoon, caught O's keeper Glyn Garner flat footed at his near post with a 25-yard strike that kept low and flew in to the net.
The introduction of Orient substitute Shane Tudor sparked the home side and Jabo Ibehre should have levelled when Daryl McMahon put him through.
At the other end Ferguson had another shot from distance as Wrexham tried to wrap things up - but this time Garner saved comfortably.
The home side finished strongly as Easton fired over and then Alexander sent a header inches wide from Tudor's cross.
10 minutes from time Lockwood delivered the perfect cross from the left and Alexander headed home his fourth goal in three games - and in stoppage time the same combination almost struck again.
But this time the skipper's header flew just over that bar.