Leeds teenager Byram turned home his fourth goal of the season early in the second half to cancel out Dwight Gayle's stunning opener for an impressive Peterborough side aiming to climb out of the bottom three.
But although Leeds pushed the visitors back for long periods after the break they were thwarted at every turn by a resolute defence in their bid to close to within three points of the top six.
Leeds substitute Jason Pearce headed agonisingly wide in time added on, but Peterborough also carved out plenty of chances with Leeds goalkeeper Paddy Kenny pulling off superb saves to deny Lee Tomlin and livewire striker Gayle in either half.
Home midfielder Michael Tonge drilled in an early low shot from outside the penalty area that tested Posh goalkeeper Robert Olejnik, but the visitors soon caused problems and striker Gayle rattled the bar in the 13th minute with a well-struck volley.
Leeds failed to heed the warning and Gayle's electrifying pace created a goal out of nothing for Peterborough in the 15th minute.
From a virtual standing start the Posh forward skipped round defender Lee Peltier on the left edge of the area and slammed a rising left-footed shot into keeper Kenny's top right-hand corner.
Peterborough striker Lee Tomlin was next to showcase his talent, swerving inside Luke Varney and then Tonge before firing in a 25-yard effort that had Kenny at full stretch to turn the ball away for a corner.
Leeds centre-half Tom Lees was forced off injured and replaced by Pearce in the 25th minute before Leeds picked up the pace.
Defender Stephen Warnock's angled volley flashed inches wide and Pearce should have equalised instead of heading Ross McCormack's pinpoint cross over the crossbar soon after.
El-Hadji Diouf replaced a struggling Steve Morison during the interval, but most of Elland Road held its breath when Gayle went head-to-head with Peltier again for a ball played over the top only to fire wide after winning the race.
Varney headed Byram's cross straight at Olejnik and Warnock's free-kick off target as Leeds went in search of an equaliser, which arrived courtesy of Byram, who guided Warnock's corner home at the far post in the 57th minute.
The game then flowed from end to end as Leeds probed for a second with Peterborough looking to counter and Warnock was relieved to be handed only a yellow card by referee Simon Hooper for bringing down Kane Ferdinand, who would have been clear on goal.
Another sweeping Leeds move ended with Diouf's cross scrambled for a corner before Gayle forced Kenny into a brilliant flying save to keep out a thumping right-footed drive.
Tomlin was hauled down on the edge of the area and clipped the subsequent free-kick just over and at the other end Varney twisted and turned in the box before shooting wide.
Leeds looked to have snatched it when Green crossed to the far post in stoppage time, but Pearce headed agonisingly wide.