The Italian's progress was relatively routine although he looked tetchy at times as he twice got involved in arguments with umpire Marijana Veljovic.
Sinner, who has dropped only one set so far, is the first Italian man to reach the quarter-finals twice at Wimbledon.
He has never gone beyond the quarter-finals at any of the four Grand Slams but now has a great opportunity to break new ground with 92nd-ranked Russian Roman Safiullin up next.
Sinner's explosive groundstrokes and much-improved serve are paying dividends on the Wimbledon lawns -- with former Australian player Darren Cahill's influence paying off.
He has worked with Cahill this summer and says the best advice he has been given is to be 'friendly with the grass'
"This is a quote that Darren gave me," Sinner said. "You'll have chances but sometimes there are some bad bounces or you can get unlucky a couple of times because your opponent is serving good. You have to always have the right mindset.
"Also sometimes trying to smile because you also have to enjoy to play here."
Galan, ranked 85th in the world, was solid throughout the opening set and Sinner looked rattled in the tiebreak when forced to replay a point after winning a Hawkeye challenge.
He did not allow his debate with Veljovic to distract him too long though as he took the opening set.
Sinner's mood was hardly helped as he dropped serve at the start of the second set and then squandered five break points in the next game, again having words with the match official.
Four more break points went begging in Galan's next service game but Sinner did finally get back on level terms for 4-4 and broke serve again to move two sets clear.
The third set was more straightforward as the Italian made it through to the second week.
"Today was a tough day. I was not feeling great on the court but I managed to win the most important points," he said. "Usually I'm calm but sometimes it happens, there were some tough calls on my side."
