Town twice fell behind to goals from Connor Wilkinson and Danny Johnson, but replies from Dan Jones and Josh McPake brought them back into the game Jack Muldoon also missing a penalty for the home side.
There was one change for Town, with Jon Stead coming in for the injured Mark Beck for his fourth start of the season.
And with lockdown restrictions having eased on Monday the majority of the Town line-up were sporting evidence of a pre-game trip to the barbers.
It was the visitors who threatened first when Dan Kemp whipped a dangerous free-kick across the Town box with James Belshaw and his defence relieved to see the ball fall to safety.
Will Smith received an early yellow card when he brought down the threating Kemp on 15 minutes.
It was Town who seized the initiative after this though. After an incisive build-up top scorer Jack Muldoon got away a shot from the edge of the area that was dragged narrowly wide.
The returning Stead had the best of the opportunities to give Town the lead when he put over the bar from the edge of the six yard box after 20 minutes.
Stead then turned provider when his clever pull-back was met by the advancing Lloyd Kerry who just failed to control his shot.
Kerry then saw his shot from distance comfortably dealt with by Lawrence Vigouroux and Connor Hall volleyed wide from the edge of the area, with Town in command at the half.
As in recent weeks the good work of the first half was undone immediately after the restart. Wilkinson, having carried the ball from distance let fly with a speculative effort that Belshaw would not relish watching back, the ball squirming through his grasp to give the visitors the lead.
However, when Josh McPake was fouled in a dangerous position Jones headed home from George Thomson's free-kick for what was Town's first goal against the O's in four meetings.
Having brought themselves back into the game after trailing for a matter of minutes Town were their own worst enemy when Jones's header back to Belshaw was intercepted by Johnson who pounced to restore the lead.
And they had further cause to kick themselves after 70 minutes when Jack Muldoon earned a penalty after being brought down. The top scorer stepped up but saw his spot kick turned aside by Vigouroux.
Town picked themselves up off the floor once again though when Hall's pass freed up McPake and the on-loan Scot fired home to level once again.
Entering the final 10 minutes Weaver's side sense the points were there to be had and Vigouroux was forced into a double save, firstly from Thomson and then the lively McPake.
With the clock ticking down Kemp had Town hearts in their mouths when his powerful header grazed Belshaw's bar but that was the last of the action as a wild second half was brought to a close with points shared.
Town: Belshaw, Burrell, Jones, Smith, Hall, Thomson, Falkingham, Kerry, McPake, Stead, Muldoon.
Subs not used: Cracknell, Fallowfield, Francis, Williams, Andrews.
Goals: Jones (58), McPake (75)
Booked: Smith (15)
Leyton Orient: Vigoroux, Ling, Happe, McAnuff (Freeman, 72), Clay, Wilkinson (Dennis, 78), Kemp, Brophy, Akinola, Kyprianou (Abrahams, 86), Johnson.
Subs not used: Sargeant, Widdowson, Coulson, Turley.
Goals: Wilkinson (46), Johnson (63)
Booked: Johnson (55), Wilkinson (56)
Referee: Gareth Rhodes.