Rudd held off Agut after a mid-afternoon thunderstorm delayed play for just over an hour.
"It was a tough battle. The first two sets were 2 hours and 20 minutes," Rudd said. "Good intensity and I was feeling it a bit in the legs. Luckily, there was some rain, which gave me some time to breath out and regain some energy. I can thank the weather gods today, giving me some extra time and energy."
"That was a very convincing win," Auger-Aliassime said. "I served really well but I felt like I was also putting a lot of pressure on him at every opportunity that I had."
Gael Monfils, the No. 17 seed from France, retired early in the second set of his match against British qualifier Jack Draper. Monfils, who trailed 6-2, 0-2, received treatment on his right ankle area but couldn't continue.