Russell Slade's Os would have felt hard done by after conceding a controversial first-half goal to an offside-looking Britt Assombalonga, but the visitors dug in and Moses Odubajo's second-half strike dragged them level going into Tuesday's second leg in the capital.
It was Orient who threatened first in the game and with only two minutes on the clock, David Mooney - who was flagged offside - just failed to connect with Kevin Lisbie's ball, which flashed across the face of the goal.
But Darren Ferguson's Posh side wasted little time in opening the scoring through Assombalonga's 33rd goal of the season.
Assombalonga (16), who looked to be in an offside position, headed in Mark Little's powerful centre after Orient keeper Jamie Jones had failed to hold Danny Swanson's long-range shot.
And Assombalonga should have doubled the hosts' lead just five minutes into the second half after John Lundstram gave the ball away.
Michael Bostwick was quickly on to Lundstram's loose pass, before clipping the ball into the path of Assombalonga, but the striker's first touch sent him wide and when he finally got his short away, it was easily held by Jones.
Peterborough maintained the upper hand and Tommy Rowe was next to go close when his cultured effort from the edge of the box drifted just wide of Jones' left-hand post.
But Orient rallied and grabbed a deserved leveller on 72 minutes through Moses Odubajo's close-range tap-in.
Sub Shaun Batt headed down Dean Cox's forward ball and after Bobby Olejnik could only parry Mooney's shot into the path of the highly-rated Odubajo, he converted from an acute angle to level the tie.
Olejnik then had to be alert to dive down at his near post to stop Cox's low drive as Orient threatened to take the lead.
Cox was allowed too much space again with 10 minutes to go, but Olejnik was equal to his powerful shot before Mooney was flagged offside with his follow-up effort.
With time ticking away, it was Orient who continued to have the better of the play and Odubajo almost grabbed his second of an entertaining game but he lashed a shot wide of Olejnik's upright.
Jones flapped at a late free-kick to give Orient a scare and the resulting corner was headed into the side-netting, but Slade's men held firm to earn a draw.