Top seed Isner broke Sweeting's serve just twice as he improved on his first-round performance, which was laboured due to the humidity in Florida.
"Whenever, in the past, when I've played tournaments in humidity I've always struggled because I hadn't acclimated yet," Isner said. "But tonight I felt so much better than I did yesterday.
"I'm still not all the way there. I could feel better out there but it was so much better."
The world number 11 will now meet eighth seed Bernard Tomic for a place in the semis after the Australian saw off American qualifier Tim Smyczek 6-1 7-5.
Tomic's compatriot Marinko Matosevic produced the shock of the day with a 4-6 6-1 6-3 win over number six seed Alex Bogomolov Jr.
It will be Matosevic's first appearance in an ATP tour quarter-final, where he will face 2010 Delray Beach champion Ernests Gulbis.
"It was a tough match and I'm happy to get through and make my first quarterfinal," said Matosevic. "I knew his game pretty well and he reaches a lot of balls, so I just had to keep staying aggressive.
"I played the big points well."
Gulbis beat Steve Darcis of Belgium 6-7(6) 6-1 6-3 to advance, although it was far from plain sailing for the Latvian.
"Today was tough because I couldn't get my timing right in the first set," said Gulbis. "From one side it was much easier to get your serve going than the other side.
"I didn't play my best tennis today but I'm happy I won the match because I was fighting."