Andress, 27, returns to Sandy Park on a one-year deal after two seasons at The Stoop and is the seventh new recruit signed by Exeter director of rugby Rob Baxter.
"Last season he played in some big games for Harlequins and he's a guy I am comfortable working with and I think he can push on again," said Baxter.
"He's more of a modern prop who does more than just the set-pieces, he gets around and makes tackles and likes to carry the ball."
Andress and fellow new front row recruit Craig Mitchell, the Wales international, will compete with current squad members Chris Budgen, Hoani Tui, Brett Sturgess and Ben Moon.
"I think he's a good signing for us and I think there is going to be a lot of competition in the squad. I like the look of the front five, I think it's an area we're going to have a lot of strength," said Baxter.
"I'm delighted with how Chris and Hoani played this year. The way Hoani played on Saturday against an England prop [Wasps loosehead Tim Payne] was fantastic.
"We pretty much dominated the scrums on our put in, so that shows Craig Mitchell and John Andress have got a big job on their hands to get in the 23 and that's fantastic for us.
"We have now got six really good props and that's a great position to be in going forward."
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