Radwanska, who won the title in 2008, struggled throughout the match at the Wimbledon warm-up tournament.
Although Radwanska tied it 5-5 when Hampton served for the first set and held a set point at 6-5, she was overwhelmed in the tiebreaker and failed to rally in the second set.
The 41st-ranked Hampton maintained her composure when No. 4 Radwanska rallied late in the first set.
"I was a little rattled, yeah," Hampton said. "Last time we played something similar happened, so I was like, `I'm not going to crack. I'm going to hold on and get to a tiebreak."
Radwanska said she's still adjusting to the grass after the clay-court season.
"Playing her in the first match on the grass, I think it was tough," Radwanska said. "The fact that she played three matches before helped her a little bit.
"I was a little bit scared to move. It wasn't slippery. Maybe a little bit in the back, but, yeah, scared. You have to be careful."
Also advancing were 2012 runner-up Angelique Kerber, who defeated Sorana Cirstea 6-4, 6-4, and fourth-seeded Petra Kvitova, who stopped Monica Niculescu 6-4, 6-1. Fifth-seeded Caroline Wozniacki advanced after leading 6-2, 2-2 when defending champion Tamira Paszek retired with a left-thigh injury.
Wozniacki faced a potentially difficult match against an opponent who not only won the Eastbourne title last year but then beat her in the first round at Wimbledon.
Wozniacki recently went 1-5 in clay-court matches, so she's looking forward to playing on grass.
"I'm feeling great," Wozniacki said. "I love the grass. It's so much fun to play on. The clay season is over, which I'm happy about. I think I played a really good first set. I felt really good and comfortable out there."
Wozniacki's boyfriend, golfer Rory McIlroy, attended the match. He finished 14 over at the U.S. Open on Sunday.