The Astana rider attacked from a small group of breakaway riders 1.5km from the finish in the final punchy climb, holding off France's Tony Gallopin and Italian Davide Formolo, who were second and third respectively five seconds behind.
Italian Valerio Conti finished in the main pack, 1:07 off the pace, to retain the overall leader's pink jersey.
The overall favourites had an uneventful day in the peloton.
"It was a really important day for me today. It took a long time to get my first victory in a three-week race," said Bilbao.
"This was only possible with a big team like Astana. Thank you to all of the team."
Slovenian Primoz Roglic, one of the top favourites, lost one of his domestiques for the mountain stages when Laurens De Plus of Belgium abandoned after being sick for a few days.
Saturday's eighth stage is a 235-km ride from Tortoreto Lido to Pesaro ahead of a key time trial on Sunday.
(Writing by Julien Pretot; Editing by Keith Weir and Mark Potter)
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