The 30-year-old will join the Chiefs in the summer, after four seasons with the Tigers.
Head coach Rob Baxter said number eight Waldrom will boost Exeter's options in the particularly specialist role.
Waldrom will likely compete with England Saxons new boy Dave Ewers in Exeter's ranks, and fill the gap left by stalwart Richard Baxter's retirement last summer.
Exeter boss Baxter believes four-cap England international Waldrom is "genuinely excited" about the move, and tipped him to thrive at Sandy Park.
"We've had a good look at him and done some pretty detailed analysis on him," said Baxter.
"What we've seen is that he's not only a very good rugby player, but he's playing well and he is someone I know will add real top-end quality to the squad.
"As we look to move forward as a club, I think it's vital that we have players like him coming into the squad.
"Although he has been around for what seems a very long time, he's still only 30 and we feel his best years are probably still to come."
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