World number 14 Goffin was edged out in a tight second set tie-break before fighting back to earn a 6-4 6-7 (7/9) 6-2 win and a last-eight meeting with Ivan Dodig.
Home favourite Dodig, granted a wild card for the event, came from behind to claim a 3-6 7-6 (8/6) 6-2 win over Italy's Marco Cecchinato.
Fourth seed Paolo Lorenzi also reached the last eight with a straight-forward 7-5 6-3 win over British number three Aljaz Bedene.
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Bedene had impressed in reaching the third round of Wimbledon earlier this month but was broken three times by world number 34 Lorenzi.
Next up for Lorenzi is Jiri Vesely, after the eighth seed saw off Radu Albot 7-6 (7/3) 6-2.
At the Swedish Open in Bastad, top seed Pablo Carreno Busta was forced to retire midway through his match with Andrey Kuznetsov.
Carreno Busta took the first set 6-4 but lost the second in a tight tie-break and was 3-0 down in the decider when he quit with an apparent abdominal complaint.
Third seed Pablo Cuevas was another seed to exit the tournament as he was beaten 6-4 6-3 by world number 100 Henri Laaksonen.

John Isner qualifié pour les demi-finales du tournoi de Rome

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Seventh seed Diego Schwartzman beat Argentinian compatriot Renzo Olivo 6-4 6-1 and eighth seed David Ferrer saw off Dustin Brown 6-2 6-3.
Big-serving duo John Isner and Ivo Karlovic, the top two seeds, had little trouble in getting through to the Dell Technologies Hall of Fame Open in Newport.
Isner, the number one seed, dropped just six games in a 6-2 6-4 win against Sam Groth, as he sent 13 aces flying down.
Karlovic was even more prolific on his serve, recording 16 aces as he beat American Denis Kudla 6-4 6-4.
Third seed Adrian Mannarino is out, though, as he suffered a shock 7-6 (7/5) 7-5 defeat to world 185 Tobias Kamke.
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