Coric, who defeated Roger Federer in last year’s final, saved seven of the nine break points he faced and took his match point after a 26-shot rally to beat Sousa in just under 3 hours.
Khachanov, ranked a career-high No. 9, hit 11 aces and converted two of his seven break chances to win in 1 hour, 43 minutes. The Russian will next face an Italian, Stuttgart Open champion Matteo Berrettini or Andreas Seppi.
Nine-time champion Federer also returns Thursday against Jo-Wilfried Tsonga of France. Tsonga won their only previous match on grass, at Wimbledon in 2011.