The hosts, who had never previously reached this last-four stage, held on at the end to remain on track for some club history.
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There was a little let-off for Exeter when scrum-half Will Chudley appeared to kick the head of Joe Launchbury without sanction, yet just moments later Dan Robson found the chalk from close range thanks to some fine interplay from the forwards.
Exeter were rewarded for their attacking play on the brink of half time when referee Greg Garner lost patience with the visitors' defence and awarded a penalty try.
Garner again found the sticks from the follow-up, and worse was to come for Wasps as Carlo Festuccia was shown a yellow card by Garner.
Kiwi fly-half Jimmy Gopperth dragged Wasps back within three points of the Chiefs with his first successful pen of the afternoon, just moments before Dave Ewers used every inch to reach over and open the hosts' advantage back up.
A couple of straightforward Gopperth kicks ensured that it was not quite all over heading into the final stages of the match, but any home nerves were finally settled when another penalty try was awarded in the dying stages, allowing Steenson to blast over once more.
That proved to be it in terms of the scoring, meaning a first final appearance for Exeter to look forward to next week.