American Sweeting, who claimed his only Tour title to date in Houston last year, saved two set points in the second set before going on to seal his progress via a tie-break.
Next up Sweeting will face either Croatian eighth seed Ivo Karlovic or fellow American Bobby Reynolds.
Another American, Michael Russell, rallied to beat Daniel Gimeno-Traver 2-6 7-5 6-1.
A set and a break down, Russell broke back as the Spaniard served for the match, then broke again before racing through the third set.
Russell will next face top seed Mardy Fish.
Horacio Zeballos beat Adrian Ungur 6-3 6-3 to advance to a second-round meeting with second seed John Isner.
There were also wins for Igor Kunitsyn, Tatsuma Ito and Paolo Lorenzi.