Eighteen-year-old fly-half George Ford kicked four crucial penalties - three of them in the second half - to help earn the win
The Tigers try was an opportunist effort in the first half by lock Graham Kitchener
Bath had romped through the pool stages, scoring almost seven tries a game, but found themselves locked in a typically hard-fought contest with their old rivals.
Now they face an uphill task to climb into the Aviva Premiership top six to qualify for the Heineken Cup.
Their own tyro fly-half, Tom Heathcote, kicked three penalties and converted a try by hooker Lee Mears.
Having announced an unchanged team from last weekend's 36-17 win over Worcester, Bath had been forced to make a late change when number 8 Simon Taylor pulled out. Carl Fearns stepped up from the bench for his first start since September.
The Tigers, already without five of the England match day squad, could have started Toby Flood and Thomas Waldrom when they were released from international duty in midweek but kept faith with the majority of players who had steered them to the semi-final.
They included fly-half Ford, still a week away from his 19th birthday.
It was a poignant occasion for ex-Leicester favourite Lewis Moody, forced to announce his retirement on Tuesday with a chronic shoulder problems and consigned to media duties. He had made just 18 appearances for Bath.
Bath made most of the running in a frenetic first quarter of an hour but a combination of alert Leicester defence and the odd handling error meant the game remained scoreless until Ford chipped over a penalty.
already shown maturity beyond his years with a series of superb kicks out of hand and it was his finely judged Garryowen that forced the penalty from ex-Tiger Sam Vesty.
Almost immediately Ford then unleashed another inch-perfect 60-metre kick to the corner. Bath fumbled and Kitchener pounced for a try which went unconverted.
Leicester had looked the hungrier side to that point but Bath were suddenly stung into action and Heathcote kicked a 30-metre penalty to cut the deficit to 8-3.
On the half hour, they took the lead through Mears after Matt Carraro carved through the Leicester midfield.
The move was carried on through Tom Biggs and Fearns before Olly Barkley threw a long pass to the Lions hooker. Heathcote converted from touch with the help of both posts.
The home side could have scored again when Matt Banahan chased his own kick to the posts, only for Ford to scoop the ball up in the giant wing's shadow and clear to touch.
The half-time break provided only temporary respite as Bath went on the attack from the break, helped by a miscued kick from Scott Hamilton.
Biggs beat five defenders on a mazy run to launch a series of attacks from which Ben Skirving nearly forced his way over.
After just five minutes of the half, Director of Rugby Richard Cockerill had seen enough and changed his props, bringing on Marcos Ayerza and Logovii Mulipola.
That paid off at the next scrum as Bath were penalised and Ford kicked the penalty.
That was immediately cancelled out as Heathcote put his side back in front at 13-11 with a fine touchline kick but Ford replied just as quickly with his third penalty to restore Leicester's one point advantage.
A great pick-up by Skirving off a retreating scrum provided the impetus for another Bath barrage.
They had to be content with a third Heathcote penalty to edge them back in front at 16-14 but he was wide with a long-range effort on 65 minutes.
Leicester showed they were far from out of it as Steve Mafi broke away from a ruck, forcing Bath to scramble back desperately until referee Dave Pearson awarded a relieving penalty.
Pearson was not so popular when he awarded Bath a penalty at a collapsed scrum only then to reverse it on the advice of his touchjudge. Ford's kick was spot on again, with just nine minutes remaining.
Heathcote was quickly given another long-range opportunity after Banahan and Cuthbert launched a counter-attack but could not find the direction.
Instead it was the Tigers who almost scored again, Horacio Aguila fielding Michael Claassens' high kick and speeding down the left touch line before being forced into touch by Cuthbert.
But the visitors had done enough to earn a place in the final at Worcester on March 18 when they will face either Northampton or Scarlets.
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