Henman is trying to become the first player to win consecutive titles in Basel since Stefan Edberg did so in 1985-86 and the second to win this event three times.
"I was pretty lucky today," he said. "(Escude) was up a set and a break, it didn't look pretty.
"I wasn't playing my best tennis and he was playing extremely well. I wasn't getting frustrated, I was hanging in there. It's amazing how quick a match can turn."
The 28-year-old Englishman, who became a father for the first time last Saturday, has played sporadically since losing in the third round of the U.S. Open in August.
"It's a good thing to have played a difficult match and to have come through it," he said. "I still think I need to improve. It's probably a matter of match practice."
Sixth seed David Nalbandian of Argentina will be Henman's quarterfinal foe Friday. Nalbandian had a 6-2, 3-0 lead against Agustin Calleri, before his countryman retired with right shoulder tendinitis.
It is the second time since his surprise runner-up effort at Wimbledon that Nalbandian has won two matches in a row. He was a quarterfinalist at the Tennis Masters event in Canada.
"It's very nice for me to be in the quarterfinals," Nalbandian said. "My game was very solid today. I knew he had some shoulder problems but I forgot it before the match. I remembered it again when he was starting to serve slower."
Second seed Juan Carlos Ferrero of Spain dispatched French qualifier Nicolas Thomann, 6-4, 6-2. Ferrero is fourth on the ATP Champions Race and is close to qualifying for the elite eight-man field for the Tennis Masters Cup in Shanghai.
"It's pretty close, everyone has to play well this week," said Ferrero, the French Open runner-up. "It's very good for me to be in the quarterfinals, I'm closer than last week. If I win tomorrow my chances are good. The Tennis Masters Cup is a big goal for me."
Ferrero is 4-0 against his quarterfinal opponent, countryman Felix Mantilla, who was a 6-1, 7-6 (7-2) winner over Switzerland's Ivo Heuberger.
"I'm very satisfied with the way I'm playing," Mantilla said. "I'm playing with more motivation than earlier. Today I played aggressive on his forehand, which paid off.
"It's very important for me to be in the quarterfinals in Basel. Tomorrow I will do my best to go even further."
Third seed Roger Federer of Switzerland beat Alexander Waske of Germany, 6-3, 7-6 (7-1). He will play fifth seed Andy Roddick in the quarterfinals Friday.
Roddick, who had a four-match losing streak heading into Switzerland, defeated James Blake, 7-6 (8-6), 6-3.
Celebrity Breakups and divorces of 2014 [PHOTOS]