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The substitute winger gathered a long, perfectly weighted skip pass from skipper and fly-half Aaron Cruden to cross the line unopposed at Newlands stadium in Cape Town.
Although full-back Damian McKenzie failed to convert — his lone failure in five shots at goal — the north island outfit survived some late pressure for a deserved if dour victory.
The reward for the Chiefs is an away semi-final against fellow New Zealanders and record seven-time champions Canterbury Crusaders in Christchurch next weekend.
Golden Lions of South Africa host defending champions Wellington Hurricanes of New Zealand in the other semi-final, which is a repeat of the 2016 final.
Crusaders beat Otago Highlanders 17-0 in a south island derby, Lions edged fellow South Africans Coastal Sharks 23-21 and Hurricanes blew away ACT Brumbies 35-16 in Australia.
“It was not the prettiest of matches,” conceded Cruden, “but we would much rather win an ugly game than lose a pretty one.
“I am proud of the boys for a massive effort — they put their bodies on the line from the first minute to the last.
“We are a team of warriors with a clear game plan that we try to execute whenever and wherever we play.
“The Crusaders in Christchurch is going to be a tough test, but we can beat any Super Rugby team anywhere on our day.”
Stormers skipper and flanker Siya Kolisi lamented some basic errors in a rematch of a 2016 quarter-final won 60-21 by the Chiefs in Cape Town.
“We did not lose to the Chiefs — we lost to ourselves. A few individual errors cost us the match.
“I am proud of the youngsters in the team, who raised their hands and did not stand back against a great Chiefs team.”
The first half developed into a penalty-kicking contest between rivals full-backs McKenzie and SP Marais, which the New Zealander won 3-1 for a 9-3 half-time advantage.
Stormers cut the deficit to a single point on 45 minutes when Kolisi dived over in the corner after sustained pressure.
McKenzie slotted his fourth penalty before Chiefs flanker Sam Cane was sin-binned on the hour having dangerously tackled winger Dillyn Leyds.
However, the numerical advantage yielded only three points for the South Africans off a second Marais penalty.
Twice champions Chiefs retained a 12-11 lead for 13 minutes before Stevenson scored one of the softest tries of his career, and one of the most important.
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