The world No. 7 and top-seeded Nishikori improved on his career-best start to an ATP season as he wrapped up victory in 1 hour and 15 minutes against lucky loser Gulbis on Centre Court on Thursday.
The Japanese star is now 10-1 for the season, having lost only to Novak Djokovic in the quarterfinals of the Australian Open last month.
Nishikori will face Marton Fucsovics for a place in the semifinals after the Hungarian world No. 38 defeated Georgia's No. 9 seed, Nikoloz Basilashvili, 7-6 (1), 6-1 in their last clash of 16. Fucsovics was runner-up at the Sofia Open last week.
Neither man faced a single break point in the first 11 games, but Tsonga -- who won this tournament in 2017 -- started to increase the pressure at 6-5 in the first set, twice earning break and set points on Griekspoor's serve. The Dutchman saved both set points and was able to hold serve to take the opening set to a tiebreak, where he saved two more set points, but Tsonga converted on his fourth attempt, sealing the set after 54 minutes.
The second set was a more straight-forward affair for the Frenchman, as a rampant Tsonga broke Griekspoor three times en route to a 6-1 set victory in just 23 minutes.
"At the start I played well, but he played well too," Tsonga said afterward. "I need to keep holding my serve and have the patience to wait for a chance. I was not surprised by his level of play ... stayed at the same level in the second set and he dropped a little bit. That was the difference."
The 2017 champion Tsonga next will face No. 5 seed Daniil Medvedev, who also lifted a trophy last weekend when he defeated Fucsovics in Sofia. Medvedev earned his place in the quarterfinals by overcoming Spain's Fernando Verdasco 6-2, 7-5 in 95 minutes on Thursday.
"Fernando is a really great player and conditions are slow here, so to find a tactic to win points was difficult," Medvedev said. "My goal was just to put the ball back in the court, and I'm happy to have won."