Neil Warnock's men surrendered a 1-0 lead in the space of four second-half minutes on Tuesday night, with the veteran boss particularly scathing of Becchio and his fellow forwards for their role in the loss.
Becchio is no stranger to criticism from his manager either, having had his fitness called into question in pre-season, but he continued to deliver answers today with a goal in each half.
Neither was the most taxing of finishes, with an abject and so-far-pointless Peterborough docile in defence, allowing first Luke Varney and then Ross McCormack to pick out the Argentinean.
And, although Michael Bostwick pulled one back for Peterborough with 17 minutes remaining, Darren Ferguson's side were unable to prevent themselves losing their opening three games of the season for the first time since 1972.
In truth, the writing had been on the wall for Peterborough from the opening minutes as, with the words of their manager no doubt ringing in their ears, the away side's front three combined to give them a seventh-minute lead.
McCormack had already had a fizzing free-kick clawed out by Robert Olejnik when the goal came, and the Scotland international was at the heart of it.
His pass set Varney free and after looking up he squared for the waiting Becchio who did the rest from 10 yards, although Olejnik could have done more with the shot.
With Rodolph Austin patrolling the midfield with far more discipline that he showed at Blackpool, Leeds looked comfortable at the back, although they took a break from that when Shane Brisley got free and flicked an effort in that Paddy Kenny did well to turn away.
But Leeds looked crisper in attack than their hosts and McCormack went close with a 35th-minute header after Lee Peltier's run and cross, although that was it as far as chances when in the first half.
No doubt pepped up by Ferguson in the dressing rooms, Peterborough were livelier after the restart, with Lee Tomlin looking to take advantage of the wet surface by drilling in from 25 yards, only to find Kenny equal to it.
But their new-found enthusiasm lasted just five minutes as Becchio grabbed a second, heading in one of the easiest goals he will ever score after McCormack picked him out, in space, with a pinpoint cross.
Tomlin refused to give up the fight for Peterborough, drawing a save out of Kenny with a header, and his spirit drew more out of his team-mates as Bostwick seized on a poor clearance from David Norris and lashed a strike beyond Kenny from the edge of the box.
Leeds were naturally nervous afterwards, with Warnock withdrawing flair players McCormack and El Hadji Diouf, although one of the defensive reinforcements sent on, Sam Byram, should have done better when a chance fell to him inside the box.
But Peterborough's fire had gone out by that point at Leeds held on to secure a 500th away win in league football.