The two try salvo came after Jones, appointed earlier this month, said every player would start with a clean slate ahead of the Six Nations in February.
"I still very much want to play for England. I'm 28, and I like the fight of it. I think the day I'm not bothered about England or Saracens is the day I'll finish," Ashton was quoted as saying by British media on Sunday.
Saracens director of rugby Mark McCall was full of praise for the winger, who had gone six matches without a try.
"We're all delighted that Ashy scored because it's been quite draining listening to him," the Irishman joked.
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"He's one of those great wingers who judges everything on whether he scores a try or not, like a good centre-forward. And when he doesn't, he's not satisfied."
(Reporting by Shravanth Vijayakumar in Bengaluru. Editing by Patrick Johnston)
