The Tykes held the lead for much of the second half but were undone by a Ross Broadfoot penalty in the 77th minute, and Tigers stand-in captain Louis Deacon secured the win with a try four minutes into injury time.
The loss means that Leeds are still at the bottom of the table.
Leicester made 13 changes to the side which beat Bath last weekend, leaving out experienced internationals such as Martin Corry, Julian White and Andy Goode.
Leicester started brightest, and Broadfoot went close with a long-range shot at goal before centre Matt Cornwell ignored an overlap after splitting the Tykes’ defence.
Leeds responded when Chris Murphy stole a lineout inside the Tigers 22, but when Andre Snyman gathered the ball in midfield he was hit man and ball by Cornwell, allowing Broadfoot to collect the ball and break upfield.
Leicester’s stole the ensuing line-out, and made far more use of the turnover.
British Lion Geordan Murphy hit the line at pace before flipping a brilliant one-handed pass out to Leon Lloyd.
The winger had too much pace for Tom Biggs and went over for the first score. Broadfoot failed with the conversion from wide out.
The Tykes hit straight back, and Roland de Marigny looked to be in for a certain score before being hauled down by a despairing James Hamilton.
The ball was moved right to Iain Balshaw and he manufactured a cut-out pass which enabled Snyman to force his way over, despite the attentions of Austin Healy. De Marigny’s conversion made it 7-5 to the Tykes.
De Marigny and Broadfoot both missed penalty attempts before the break, but there was still time for Tigers to take the lead when Cornwell latched onto a Broadfoot pass, after some sustained pressure, before slipping the tackle of Justin Marshall and diving over for the score five minutes into injury time. Broadfoot’s conversion made it 12-7 at half-time.
Deacon was sin-binned on 42 minutes after what looked like an accidental collision with Balshaw.
Leeds capitalised immediately, with their pack mauling ferociously and creating the chance for Murphy – Deacon’s opposite man – to crash his way through a crowd of players to score. De Marigny’s conversion put the Tykes 14-12
The Italian fly-half added a further penalty before Deacon was able to rejoin the action.
De Marigny’s fourth successful kick of the afternoon made it 20-15 to the Tykes, but the Tigers displayed all their traditional fighting qualities, and another Broadfoot penalty made it 20-18 to the home side with 11 minutes to go.
Tigers took the lead with three minutes of normal time to play after Tom Palmer failed to roll away from the tackle. The increasingly assured Broadfoot slotted a difficult kick from the 10-metre line before Deacon’s late try – converted by Broadfoot – condemned Leeds to yet another defeat.