The visitors were well in control after midfielder Andy Drury had given them a 34th-minute lead, but once goalkeeper Alex McCarthy, who has just completed a loan spell at Leeds before joining Ipswich last week, was sent off for deliberate handball in the 71st minute, Paul Jewell's side fell apart.
Defender Ibrahima Sonko handed Snodgrass a 74th-minute equaliser by playing the ball to the Leeds winger in the six-yard box, McCormack capitalised on replacement keeper Arran Lee-Barrett's indecision eight minutes later and Becchio pounced on Tommy Smith's error in the 91st minute to lift Simon Grayson's side back to within a point of the play-offs.
Jewell must be wondering what he can do to halt his side's slide down the table after seeing them crash to a fourth defeat in their last six league games.
Several hundred Leeds fans staged an angry demonstration before kick-off against chairman Ken Bates, who is currently on holiday in South Africa.
Demonstrators unfurled a large banner condemning the chairman's seven years at the club as this week's sale of skipper Jonny Howson to Norwich has fuelled their anger over a failure to invest in players.
Midfielder Howson has been replaced, temporarily, by Fabian Delph, who returned to Elland Road on loan from Aston Villa on Friday and it was the 22-year-old Leeds academy product who had the first attempt on goal, but his 25-yard shot in the 27th minute was blocked by Drury.
The visitors took a deserved lead in the 34th minute. Martin, who had been causing Leeds plenty of problems out wide, played the ball across the penalty area from the left and Drury swept the ball home into the bottom corner from 20 yards.
Neither side created any openings in the first 10 minutes of the second half and found it difficult to retain possession in the swirling rain.
Ipswich goalkeeper McCarthy did not make his first save until turning away Leeds defender Zac Thompson's cross in the 61st minute.
But Leeds were handed their first gift 10 minutes later when Ipswich were reduced to 10 men.
McCarthy inexplicably handled the ball outside his penalty area following a hopeful punt forward and three minutes later replacement keeper Lee-Barrett - striker Jason Scotland had been sacrificed - was picking the ball out of his net after the visitors had pressed the self-destruct button again.
Leeds substitute White crossed the ball from the right and Sonko, under pressure in the six-yard box, could only play the ball to Snodgrass, who duly converted.
Another howler from Ipswich, this time from Lee-Barrett, handed Leeds their second goal in the 82nd minute.
Adam Clayton crossed from the right and McCormack took advantage of the dithering Lee-Barrett's indecision to pinch the ball and roll it into an empty net, while Becchio stole in to add a third from a tight angle after Smith had failed to clear the danger in time added on.