Playing against an out-of-form Wasps side, who were effectively out of the competition even before kick-off, the Chiefs were dominant at the scrum and showed more of a cutting edge in attack.
Exeter were also excellent in defence to restrict Wasps to just one try through Mike Powell, compared to their three touchdowns from Dave Ewers, Myles Dorrian and Ally Muldowney, thanks largely to some ferocious tackling throughout as they stayed in touch with Northampton in Pool Four.
Wasps gave debuts to former Leicester winger Lee Robinson and Namibia international Heinz Koll, while the Chiefs gave a first start to teenage centre Sam Hill.
It did not take long for Exeter to open the scoring as Steenson slotted over a penalty from the middle of the pitch 40 metres out.
But Wasps quickly levelled matters as former Chief Ryan Davis converted from a similar position as the visitors were penalised for handling in a ruck.
The early stages were punctuated by stoppages, but Steenson was restored his side's lead with a simple kick in front of the posts, only for Davis to reply in kind again.
The kicking duel was broken as Exeter scored the game's first try with Pat Phibbs feeding the ball to Ewers, who powered over on the left and Steenson converted from out wide.
Davis pulled back three points with his third penalty but the Chiefs were the more potent and scored again when Phibbs broke clear and passed to allow Dorrian to touch down under the posts.
Steenson added the simple conversion before the Irishman kicked another penalty as Exeter went into half-time sitting pretty at 23-9.
The visitors tried to turn the screw even further early in the second half and came close as James Phillips knocked on just short of the line.
But they soon extended their lead again as replacement Muldowney scuttled his way over from close range, with Steenson again striking the ball superbly to convert.
This gave them a real chance of bagging a bonus point but a raft of replacements from both sides took some momentum out of the game.
The hosts gave youngster Will Robinson a debut off the bench at fly-half but they just could not breach the Chiefs' defence with Nathan Morris superbly tackled just short after an attacking line-out.
Morris and Bob Baker were then the latest to be frustrated, but the late pressure eventually told, even if it did not affect the result.
Powell showed great pace to spot a gap on the right wing and get away despite Exeter defenders closing on him from both sides, with Robinson converting the final points of the game.
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