Weston grabbed the only goal of the FA Cup second-round game at Brisbane Road when he fired home from 10 yards after a long throw from Charlie Lee fell his way.
Until then it had been a drab affair with neither team able to offer much in the creativity stakes, although Gills did have two shouts for penalties turned down by referee Phillip Gibbs
Their strongest claim came when O’s centre-back Scott Cuthbert appeared to handle a header from striker Danny Kedwell.
Kedwell had the opportunity to double his side’s lead after 57 minutes. Orient ‘keeper Ben Alnwick sliced his clearance straight to the Gills’ leading scorer who, just 12 yards out, dallied and allowed Alnwick to recover the situation.
Lee Cook almost spared the League One side’s blushes in the final minute of the tie when he crashed a shot from close range against the underside of the crossbar, but that was the only real chance offered by the disappointing home side.