Scotland international striker McCormack scored the first and last goals of an end-to-end game, with Becchio, back after a five-month injury lay-off, equalising in between times after Paddy McCarthy and Sean Scannell had put Palace in front.
McCormack's winner, his sixth of the season, came six minutes from the end and earned his side a second win of a campaign that has so far been overshadowed by off-field unrest about a perceived lack of investment from outspoken owner Ken Bates.
The situation has even prompted the former Chelsea chairman to call sections of the club's support "morons" and the bad feeling inside the ground swelled further when Dougie Freedman's Palace, sixth before kick-off, went into a deserved lead.
But first Becchio and then McCormack ensured that some of the gloom was lifted, although Palace could well consider themselves unlucky to return to London with nothing.
It had looked as though they may end up suffering a heavier defeat when Leeds stormed into an eighth-minute lead.
Charlie Taylor, a 17-year-old left-back making his first senior start, whipped in a cross from close to the halfway line, and McCormack did the rest, peeling off his marker before planting a header back across Julian Speroni.
The celebratory mood amongst the majority of the crowd only lasted three minutes, though, with Palace moving quickly to equalise through McCarthy.
A needless foul by Michael Brown on Scannell gave the Eagles a free-kick on the corner of the Leeds box, with Mile Jedinak's ball in reaching the Irishman who stabbed in at the far post.
The majority of Leeds' good work in attack was coming through Taylor and it was after he was fouled at the corner flag by Wilfred Zaha that the home side hit the post and Palace subsequently went ahead.
Ramon Nunez spotted Speroni was positioned too far back and whipped in a near-post free-kick, but, even though the Argentine scrambled and padded the ball down, possession fell to Robert Snodgrass who hit the outside of the post on the follow-up.
Not hanging around, Speroni spotted the ball and launched an attack which set Jermaine Easter free down the left.
The well-travelled frontman made light work of skipping past Paddy Kisnorbo and his pinpoint centre was turned in by Scannell who got to the ball just ahead of Darren O'Dea.
The goal proved to be the trigger for the home fans to turn on their team, and their ire was nearly increased further when Scannell's cute header crept just over the bar.
Nunez, McCormack and Andy Keogh all had half-hearted efforts at getting Leeds back on terms, but it was Palace who remained the better side, with the combative Scannell running rings around the experienced Brown in the centre of the field.
Zaha nearly made it three before the break when his lob forced Andy Lonergan into a desperate save, and the keeper was in the thick of the action again after the restart as he bundled Easter's stinging drive around the post.
Grayson threw on last season's leading scorer Becchio and new signing Mikael Forssell in an attempt to liven his side up, something Snodgrass nearly did with a curler that floated narrowly over.
Scannell then worked Lonergan with a 25-yard effort that caught everyone by surprise, before Becchio nodded in McCormack's free-kick, awarded after another Zaha foul on Taylor, with 22 minutes remaining.
Leeds had the momentum they needed now and with time running out, Forssell's flick-on found McCormack and the former Cardiff man spun McCarthy before planting an effort beyond Speroni.