Two goals in two second-half minutes from Lewis Grabban and Marc Pugh earned the home side only their second victory of a miserable season so far.
In a first period with not much to get excited about, Orient's former Bournemouth striker Michael Symes attempted an ambitious lob over David James only to see it sail wide.
At the other end Eunan O'Kane broke clear for the Cherries before taking too long on the ball and shooting straight at Allsop.
A poor game sprung into life in the 64th minute when Grabban turned home a Simon Francis corner at the back post.
Pugh sealed the win moments later as he drifted past two challenges before rounding Orient goalkeeper Ryan Allsop and slotting home from six yards.
Charlie Daniel almost topped off a perfect afternoon for the hosts only to have his curling 30-yard free kick pushed wide by Allsop.