Del Potro, who won 6-1, 6-2, will next face Paul-Henri Mathieu after the French player beat compatriot Michael Llodra 6-4, 6-3.
Del Potro is fresh from a successful Asia tour, winning the Tokyo title and reaching the Shanghai Masters final against Novak Djokovic after beating Rafael Nadal.
The Argentine, ranked fifth in the world, has not dropped a set so far in Basel, where he beat five-time champion Roger Federer for the title a year ago in the Swiss player's hometown.
Del Potro's victory in 72 minutes was his 45th win of the season, leaving him with 11 quarter-finals or better from 16 events played.
"The start was tough for me," said the 2009 US Open winner, who broke his opponent twice in each set and took victory on a fourth match point.
"I had a tough service game to start but then I gained confidence after I broke him for the first time," he said.
"I served better than yesterday and generally played a better match."
Del Potro put Baghdatis under pressure in an eight-minute sixth game in which the Cypriot saved four break points to win his first game of the match for 1-5.
Del Potro finished off the set a game later and broke in the fifth and seventh games of the second set to conclude the win.
He has already qualified for the eight-man season finals in London starting a week from Monday but would find winning a second straight title in Federer's territory a welcome boost.
"I like my level right now, I'm playing well but I'm still far behind the top players," he said.
"Their level is better than mine and I have to keep working to improve."
In other second-round results, Bulgarian eighth seed Grigor Dimitrov beat Alexandr Dolgopolov 6-3, 6-2, winning his seventh match in a row after claiming a career-first title at the weekend in Stockholm.
He meets Federer in Friday's quarter-finals.
Canadian Vasek Pospisil reached his third quarter-final of the season with a 6-3, 6-4 defeat of Ivo Karlovic.
Frenchman Edouard Roger-Vasselin put out German Tobias Kamke 7-5, 6-3 while Daniel Brands of Germany reached the last eight as he defeated Poland's Lukasz Kubot, 6-2, 6-4.