Nishikori won just 14 points in the first set and nine in the second, losing to Thiem for the first time.
Thiem, the only man to beat Rafael Nadal on clay this year, held off a short-lived Nishikori revival on Court Philippe Chatrier to set up the meeting with Zverev.
Buzarnescu, a 30-year-old Romanian with a doctorate in sports sciences who had never won a Grand Slam match before this week, downed Ukraine´s fourth seed Elina Svitolina with a shock 6-3, 7-5 victory.
Both players were bidding to reach the last-eight stage at a major for the first time in their careers but it was Zverev, who recorded a 4-6 7-6 (7-4) 2-6 6-3 6-3 win in three hours and 29 minutes.
The world No 3 will be the youngest quarter-finalist in Paris since a 20-year-old Juan Martin del Potro in 2009 and Zverev was clearly delighted with his achievement. "It's the best, a five-set match". "I'm very happy about being in the quarterfinals here going the hard way, going the long distance. showing everybody that I can play for as long as I need to". "This definitely paid off, the hours in the gym every day ..."
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Alexander Zverev jokes that it took him five sets to beat Karen Khachanov because he 'might as well stay longer for practice!'
He led 4-2 in the final set, saw off a match point and then bounced back to secure the win.
The second-set tiebreaker was scrappy, Zverev clinching it with an ace to level the match.
World number 38 Khachanov accelerated away with the third set, earning a double break after Zverev missed a chance to hit back.
Following his three-and-a-half hour match win over the Serbian, Zverev became enamoured with the distinct British accent when radio broadcaster Jonathan Pinfield asked whether Roland-Garros could be a turning point in his career.
It so easily could have been eight, with Dzumhur leading by a break on two occasions in the critical fourth set and serving for the match at 6-5 up, while also crafting a match point on the Zverev serve deep into the decider.