Hibs have started the process of appointing a new boss after thanking Stubbs for doing "a magnificent job" over his two-year spell.
Stubbs, 44, decided to return to England shortly after ending Hibs' 114-year wait to win the Scottish Cup.
"They fitted in with my ambition," Stubbs said of Rotherham.
The Englishman has signed a three-year deal with the Millers, who reached a settlement with Hibs over compensation for Stubbs and his assistant, John Doolan.
"Alan and his staff have done a magnificent job, with the pinnacle obviously being that historic Scottish Cup win for which every Hibernian supporter is thankful," said Hibs chief executive Leeann Dempster. "We wish them every success in the future
"The task now for Hibernian is to find a head coach who will lead us back into Scotland's top flight by winning the Championship this season.
"We expect high calibre candidates, and while we understand there will be a lot of speculation, as ever this club will go about its business quietly and professionally and speak when there is something meaningful to say."
Speaking for the first time since his appointment at New York Stadium, Stubbs admitted it was a "really difficult decision" to leave Hibs, but that moving to the Championship in England was "the next chapter" for his career.
Rotherham finished one place outside the relegation zone last season but Stubbs believes he can guide the team up the table to match his own ambitions and those of chairman Tony Stewart.
"I was impressed by what the chairman had to say, what his vision is for the club and the ambition," he explained.
"The club's had two seasons where they've stabilised in the Championship and we're looking to take the club forward. We want to be moving up the table and ultimately we want to be a club that is around the play-offs. That has to be the aim.
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"The plan has always been to be the best I can. I wanted to manage at the very top level. I got a great opportunity at Hibernian, they've been fantastic with me, I really enjoyed my two years there.
"I'm glad I've been able to leave on the back of a successful end to the season but now it's a new chapter and one that's very exciting for me."
Stewart insisted that, despite lengthy talks with Hibs over compensation - and Stubbs and Doolan ultimately resigning from their posts - relations between the two clubs were fine. And he confirmed that compensation will be paid to the Easter Road club.
"Alan had another year on his contract, so we did the protocol with Hibs," Stewart said. "My CEO had discussions with them to talk to Alan and that led to the appointment. We agreed a settlement with Hibs and we move forward now freely.
"We've followed Alan's career. He was on the radar months ago, the timing was right and we took the opportunity. I like the drive, the leadership.
"Football is about moving around. Hibs have been fantastic, Alan's told me how good they've been up there and it's sad they've lost a manager and Rotherham gained him. We look forward now to getting ready for next season."
Alan Stubbs says leaving Hibernian for Rotherham United was a "calculated risk" but "the right decision at the right time".