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Home advantage told in the opening throes of the last-four clash as Michele Rizzo crossed over with just five minutes played.
That proved to be Leicester's only try of the game, though, as they relied on penalties to keep track with Exeter.
Exeter were able to get a grip on the game with 20 minutes on the clock when Fetu'u Vainikolo picked off a pass in the middle of the field before racing in under the posts.
Two kicks apiece from Bell and Ceri Sweeney made sure that things remained tight heading into the half-time interval, but it was the Chiefs, three places above their opponents in the Aviva Premiership table, who showed the extra steel required following the break.
Having been caught offside straight from the restart, Sweeney added further points from the boot to put the current holders on their way.
A strong spell of pressure from Leicester culminated in a drop goal for Bell and when Max Bodilly was shown yellow for the visiting side for a professional fou,l it looked to be game on once more.
Bell was on target with the resulting penalty to ensure that there was just one point in it with a quarter of the match still left to play.
However, any hopes of a late twist were put to bed by replacement Greg Bateman, who crossed over on the 85-minute mark to secure Exeter's place in next week's final against Saracens.