Bencic, 18, romped through the deciding set to clinch the title with a 6-4 4-6 6-0 victory which further underlined her good form on grass ahead of Wimbledon which starts on Monday.
Bencic, ranked 31st in the world, reached the final of the Den Bosch event in the Netherlands at the start of the month. She will begin her Wimbledon campaign against Bulgarian Tsvetana Pironkova.
Radwanska, a former champion at Eastbourne and Wimbledon runner-up in 2012, collapsed in the final set having clawed her way back into the contest as her opponent played near flawless tennis.
"This is one of the biggest days of my life, it's a dream come true," Bencic said.
"I am so happy. It's incredible. I will never forget this."
At 18 years and 109 days, Bencic is the youngest player to win a premier-level title since Caroline Wozniacki won in New Haven almost seven years ago.
In the men's grasscourt event in Nottingham, Uzbekistan's Denis Istomin claimed his first ATP title by beating Sam Querrey of the United States 7-6(1) 7-6(6).
Istomin, ranked 92nd in the world, survived a second-set rally from Querrey who came back from 5-2 down to force a second tiebreak.
Istomin was broken twice while serving for the match at 5-2 and 5-4 and then lost five of the first six points in the tiebreak.
The Uzbek, however, won seven of the last eight points to take the trophy.
(Reporting by Toby Davis; editing by Ed Osmond)
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