Mansfield finished with a flourish and after Callum Lloyd cut the deficit late on. The home side were desperate to hear the final whistle. Earlier, it had all looked plain sailing for Martin Ling's side as they took a vice-like grip on the game, and overworked Mansfield defender Luke Dimech was forced to clear a Daryl McMahon header off the line.
Orient deservedly opened their account after 26 minutes when some fine build-up play resulted in Derek Duncan whipping in a cross from the left that Alexander fired home from close range.
Seven minutes later the Orient skipper struck again with his ninth league goal of the seaon as he headed home from six yards following Lee Steele's perfect cross.
Orient almost added a third on the stroke of half time when Michael Simpson's quickly taken free-kick from the half-way line caught Town goalkeeper Jason White off guard and he had to scrabble back to tip the ball over the bar.
Mansfield boss Carlton Palmer threw Rhys Day and Giles Coke into the action at the start of the second half, but it was Orient who should have stretched their lead as Alexander, Steele and Matt Lockwood all had chances.
As the half progressed the visitors finally found some rhythm, and with the wind at their backs they dominated the latter stages.
Lee Harrison, in the home goal, made a smart stop to deny Simon Brown 20 minutes from time - Mansfield's first real effort on target.
But Town then stepped up the pace, and Lloyd cut the deficit two minutes from time following a corner.
In a frantic finale Mansfield almost grabbed a point, but Orient defender Gaby Zakauni cleared substitute Fraser McLachlan's effort off the line, and then in stoppage time Harrison pulled off another super save to deny Brown.