Sixth seed Richard Gasquet overcame top seed Gael Monfils in an all-French quarter-final 6-7 (5/7), 6-4, 6-2.
A break in the third game of the decider proved enough for Tsonga to wrap up the win as he goes in search of a third Open 13 title, having triumphed in 2009 and 2013.
"I thought I served and returned really well today", said Kyrgios.
The 21-year-old Australian fired down 15 aces and lost just six points on serve in a ruthless display that lasted less than an hour.
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NY fans might be frustrated with such moments from their team's stars in moments that feel symbolic of their larger season. Courtney Lee had 25 points for NY , which has lost six of seven and is 12th in the Eastern Conference playoff race.
Tsonga improved to a 17-3 record on the season.
"I wanted to show him I'd be there for each shot", added Tsonga, who will now climb to eighth in the ATP world rankings.
He will next meet Jack Sock, who came through an all-American semi-final with Donald Young in a 6-4 7-6 (7/2) win.
Pouille was going for his second ATP World Tour crown after winning his maiden title at the Moselle Open in Metz last September.
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After jumping out to a 5-0 lead in the second set, Gasquet won three straight to make a match of it, but Pouille served out the match on his second opportunity, clinching with a forehand volley into the open court. "I tried my best, but he was too good".
He saved the only break point he faced in the match and won the final seven points of the second-set tiebreaker to seize the victory.
The Philippine-Belarussian duo were due to face second seeds Raven Klaasen of South Africa and Rajeev Ram of the USA in the final after they ousted Jonathan Erlich of Israel and Scott Lipsky of the United States 6-3, 3-6, 10-1.
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They've beaten Notre Dame, Virginia and Louisville this month alone, with their lone February loss coming in a close game at Duke. The Tar Heels' relive losing on Kris Jenkins' 3-pointer without being prompted and they don't want to forget it either.