Billy Twelvetrees came off the bench to replace the injured Toby Flood and spark the Tigers, who had led just 3-0 at the break, into life.
The 23-year-old England Saxon centre, who will be leaving for Gloucester at the end of the campaign, kicked 10 points and created two of his side's three tries in a win which earned them a showdown with Saracens.
It means the two teams will not meet at Twickenham for a third straight season, with Leicester having the chance to avenge last year's final defeat.
There was little for the sell-out 24,000 crowd to cheer in a lacklustre, mistake-ridden first half which ended with the hosts in front thanks to a Flood penalty.
Both defences were in form, and Bath will have been delighted with the way they repelled numerous Leicester attacks in the second quarter.
Flood missed a long penalty in the third minute before converting another four minutes later, with Bath centre Olly Barkley missing a 45-metre penalty shortly afterwards.
Leicester, wearing their new strip for next season, should have scored when winger Horacio Agulla blasted through the Bath defence but Geordan Murphy was unable to hold on to his poor pass with the line at his mercy.
The game finally burst into life in the 29th minute when Manu Tuilagi wriggled free of several Bath players. Alesana Tuilagi kicked into the 22 and felt he was blocked before Manu surged down the left only to knock on in the tackle.
Flood almost made the line before being nailed by Claassens and then Thomas Waldrom launched another attack from a free-kick.
But, again, Bath's defence not only held firm but scored a psychological point when they pushed the Tigers scrum back on their own line.
Flood hit a post with a long-range penalty right on half-time and his replacement, Twelvetrees, also missed with a 40-metre penalty seven minutes after the break.
But, after exchanging penalties with Barkley, he sparked Leicester into life with a fabulous outside break which ended with a try for centre Anthony Allen.
Twelvetrees put Agulla away down the right with a lovely floated pass and after some fierce pressure he popped up near the line to provide an inside scoring pass to Allen before slotting the conversion.
Twelvetrees extended their lead to 16-3 with his second penalty and then converted a penalty try in the 71st minute when the referee lost patience with Bath' scrum.
The departing Tiger wasn't finished there, however, as he provided the scoring pass for Agulla after a break by Tom Youngs.
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