The home side's point during the opening 40 minutes came courtesy of a try from Tom Johnson.
After the restart Exeter's Mieres kicked two penalties, while fly-half Steenson struck added a penalty-kick of his own.
Steenson, 27, then converted a try from Chris Budgen to help the Chiefs move up to fourth in the Aviva Premiership table.
Jack Wallace ran in a consolation try for Wasps two minutes from time, but Davis failed to make the conversion.
The visitors remain seventh in the standings.