The Tigers looked on course for victory via first-half tries from James Buckland, Sam Vesty and Ollie Smith.
But they found themselves 23-19 behind with minutes to go before Will Skinner saved the day to heap misery upon Leeds.
Vesty’s conversion sealed a victory that puts Leicester third in the table and Leeds rooted to the bottom.
Leicester’s position could change by the end of the weekend with Gloucester travelling to Northampton on Saturday and London Irish going to Bristol on Sunday.
Leeds had handed themselves a relegation lifeline with three wins in their previous five games before the trip to Leicester.
They drew first blood with Roland De Marigny’s second-minute penalty from the right-hand touchline.
Both sides struggled to find touch in a scrappy opening until Leeds were penalised and Austin Healey belted his kick into touch on the 22 metre line.
That set up a drive for the line from 10 metres out and Buckland touched down in the eighth minute for his first try of the season. Vesty converted to make it 7-3.
Healey’s clever kick into touch five metres from Leeds’ line kept the pressure on the visitors and the ball ran loose from the lineout.
Tigers got the ball and Healey and Hipkiss combined to send Vesty over in the left-hand corner after 16 minutes. Vesty’s conversion was wide of the posts.
De Marigny was more accurate with a penalty from just inside his own half when Tigers were caught offside after 21 minutes.
Leicester hit back with their third try following another error by Leeds at the lineout.
That led to a scrum and Healey whipped a flat pass to Smith, who spotted a gap and powered through to touch down between the posts.
Vesty’s conversion made it 19-6 after 26 minutes.
But Leicester had to survive a spell of heavy pressure just before the break.
Leeds roared back after the interval. De Marigny’s penalty reduced the arrears after 46 minutes and then Justin Marshall’s converted try after 68 minutes set up a nail-biting finish with his side trailing 19-16.
Leeds went ahead with just four minutes to go when a spell of heavy pressure on Leicester’s line ended with Tom Biggs diving over in the right-hand corner.
De Marigny added the conversion to put the visitors 23-19 ahead and leave Leicester moments away from their first defeat at home in a Premiership since January 2004.
The home side roared back with the backing of a sell-out crowd and Healey’s kick into touch five metres from the Leeds line, after the visitors had infringed, set up a catch-and-drive in the fifth minute of injury time.
Skinner grounded the ball amid a mass of bodies and a try was awarded after referee Chris White consulted the television replay.