Rashid is uncapped at test level but impressed in the recent one-day series against New Zealand. Finn has not played test cricket since July 2013.
"There are no huge surprises in the squad," National selector James Whitaker said.
"Adil Rashid's inclusion provides (captain) Alastair (Cook) and (coach) Trevor (Bayliss) with plenty of bowling options and the surface at Cardiff will dictate who makes the final eleven.
"Within the squad there are players who have plenty of experience of Ashes cricket and that knowledge will I'm sure be useful for the players who are yet to experience a home Ashes series."
Rashid, 27, was the only member of the 13-man party who was not part of the four-day training camp in Spain last weekend.
"(I'm) delighted, over the moon. I'm very excited to be part of the Ashes squad and to be just part of the team so I'm really looking forward to it," he told Sky Sports News.
"It's nice to be part of the squad and if I do get a heads up, I'll obviously try my best and if I don't, it's no drama. I'm still part of the squad and it's always a learning curve to be playing or not to play."
The first test starts in Cardiff on Wednesday.
England suffered a humiliating 5-0 whitewash in the last Ashes series in Australia.
Squad: Alastair Cook (captain), Moeen Ali, James Anderson, Gary Ballance, Ian Bell, Stuart Broad, Jos Buttler, Steven Finn, Adam Lyth, Adil Rashid, Joe Root, Ben Stokes, Mark Wood
(Reporting by Justin Palmer; editing by Ed Osmond)
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