The in-form Argentinian scored his 19th goal of the season on the hour to keep United in with a chance of making the fourth round, although a replay was the last thing either manager wanted.
Both Neil Warnock, absent through sickness, and Lee Clark have long injury lists to deal with and could have done without another game, now in the diary after Becchio's cancellation of Wade Elliott's opener.
The goals were the only real high points on an otherwise forgettable afternoon, with clear-cut chances at a premium, although Birmingham could make a case for being the better team.
Jason Pearce's downward header from a Ross McCormack free-kick rebounded off a post, before David Norris was denied by Colin Doyle, deputising for the in-demand Jack Butland, who was rested on the bench.
The running of Ravel Morrison, improving every game for City over recent weeks, was causing Leeds a number of problems, forcing them to sit deeper and, with 33 minutes gone, they were caught out as Blues took the lead.
Having tried his luck earlier, Elliott ran with the ball from the halfway line and, 25 yards out, fizzed one past Jamie Ashdown and into the top-right hand corner for a quite spectacular opener.
Leeds took a while to respond and threw on El-Hadji Diouf at half-time in a bid to do so, but were not helped by leading goalscorer Becchio getting his feet mixed up as he got clear, having to settle for a volley which Doyle dealt with.
Birmingham continued to impress at the other end and Nathan Redmond was unfortunate to see his cross-shot after a jinking run loop to safety, and the near-miss came back to haunt City as Becchio found his range on the hour, neatly slotting under Doyle after Norris showed good feet to send him clear.
A nasty-looking injury to teenager Will Packwood then held play up for five minutes, although Redmond should have made their re-adjustment easier when he burst through past Pearce, only to fire wide. That was about as good as it got from there on in, with neither side able to create the chance they needed to ensure a replay could be avoided.
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