Neil Danns had opened the scoring from close range in the closing stages of the first half, but Becchio's headed equaliser in the 81st minute was followed by a well-taken second moments later as the Whites stretched their unbeaten run to seven games - and Becchio took his goal tally for the season to 11 .
Leeds started brightly and had the ball in the net early on, although Jonny Howson seemed to be the only player in Elland Road not to notice the linesman flagging for offside as he ran off celebrating.
Palace then created a couple of half-chances as James Vaughan found some space and curled a shot over the crossbar from the edge of the box, and Darren Ambrose was then denied by the legs of Kasper Schmeichel.
The game continued to flow, but Leeds should have opened the scoring in the 38th minute when Robert Snodgrass cut the ball back for Becchio. The Argentinian striker failed to control the ball, though, and Max Gradel sent the loose ball wide.
The home side were hit by a superb counter-attacking goal before the break. Nathaniel Clyne did well to get to the byline and get the ball across. It was behind Vaughan but reached Danns at the far post and he side-footed home his fourth goal of the season.
Palace, with three wins from their last four games before today, almost doubled their advantage in first-half injury-time. Ambrose whipped a 25-yard free-kick over the wall, but Schmeichel somehow managed to tip the ball over for a corner.
Neil Kilkenny drilled in a shot which was well held by Speroni at the start of the second half, but it was the Eagles who looked the side more likely to add to their lead.
Ambrose let fly from 30 yards but Schmeichel brilliantly clawed the ball to safety, before a low, hard drive from Danns brought an unorthodox save from the Dane.
Simon Grayson's men hit back, with Alex Bruce heading wide from close in and Becchio forcing Speroni into a point-blank save with another headed attempt from a deep Snodgrass cross.
Sub Lloyd Sam brought another decent stop from Speroni before Palace came agonisingly close to sealing the points as Danns' right-footed shot came back off a post.
Sam was trying his best to get the Whites back into the game, but again the goalkeeper was equal to his powerful strike on the run.
Leeds finally got the goal they deserved with nine minutes remaining. Kilkenny's corner was flicked on by George McCartney and then Billy Paynter, and Becchio was on hand to get the final touch over the line.
And the same player snatched a second shortly afterwards, collecting the loose ball and holding off the challenge of a defender before slotting the ball past Speroni.