The hosts, who are protecting an 11-match unbeaten run, raced out of the traps but struggled in their attempts to create clear-cut opportunities.
Kevin Lisbie, who passed a late fitness test to start, was the recipient of the game's first real chance, but defender Dean Lewington produced a last-gasp challenge to avert danger as the 34-year-old was about to pull the trigger.
The O's monopolised possession but, despite their territorial dominance, Russell Slade's men fell behind on the half-hour mark.
Chelsea loanee Bamford galloped down the flank, before cutting inside and firing past goalkeeper Jamie Jones from 18 yards.
With their undefeated record under some threat, Orient responded well to the setback and could have been awarded a penalty, as David Mooney went down under pressure by Tom Flanagan but the referee shrugged appeals by the home players.
However, the hosts were soon level. Mooney wriggled down the right and crossed for Odubajo, who dispatched a curling effort past Ian McLoughlin to preserve Orient's hopes of extending their unbeaten streak.