The fifth-seeded del Potro failed to convert four match points when serving at 5-4 in the fourth set, but maintained his composure to close out the contest with his first opportunity in the tiebreaker that followed.
The only fourth-round contest to be carried over to Tuesday lasted 4 hours, 24 minutes, making it the longest men's singles match of the tournament.
It is the first time since he reached the Australian Open and French Open quarterfinals in 2012 that del Potro has qualified for the last eight at consecutive Grand Slams.
He reached the semifinals at the All England Club in 2013, but four wrist surgeries -- three on the left -- almost made him retire.
The Argentine next faces two-time champion Rafael Nadal, who has reached the Wimbledon quarterfinals for the first time since 2011.