They stay hot on the tails of Wycombe and Grimsby by inflicting more gloom on the Pilgrims who have now won only one of their last nine league games.
Both sides showed plenty of lively tenacity in the opening exchanges, but the closest to an early goal was a miscued shot by home skipper Stewart Talbot which bobbled through to Orient keeper Glyn Garner in the 10th minute.
The Londoners replied with a couple of half chances created by leading scorer Alexander. But neither troubled home keeper Conrad Logan.
Back at the other end Boston went close again on 24 minutes when David Galbraith got on the end of a Brad Maylett cross but fired wide.
Three minutes later Simon Rusk flashed a 25-yard free-kick a yard wide of an Orient post and moments later a low Francis Green cross eluded everyone as Orient survived again.
The visitors hit back and almost grabbed a 35th minute lead when Shane Tudor accepted a quick free-kick from Matt Lockwood but drilled across the face of goal and wide.
However, Orient went ahead in the 43rd minute when Ibehre raced on to a through pass by Justin Miller, shook off Paul Ellender and fired past Logan.
Boston pressed at the start of the second half and drew level on 50 minutes when Talbot netted his first goal of the season. He swivelled to fire home after keeper Garner had weakly punched out a Galbraith corner.
Orient hit back and Lockwood tested Logan with a 58th minute 20-yard free-kick. But two minutes later the visitors regained the lead when Alexander netted from close range after Logan had produced two wonder saves in a penalty area pinball scramble.
The visitors almost made it 3-1 on 65 minutes when Tudor fired over a curling free-kick, but Logan punched clear in decisive fashion.
Boston exerted a lot of late pressure but found the Orient rearguard in resolute form. However, Lawrie Dudfield almost levelled matters four minutes from time when bursting through only to fire inches wide.