Robins, 43, had transformed Coventry's season after being installed as Andy Thorn's successor in September, steering them from a potential relegation battle in npower League One to the fringes of the play-offs.
But Huddersfield, who sacked Simon Grayson three weeks ago after going 12 games without a win, were granted permission to approach Robins on Tuesday after Coventry had rejected an initial approach.
Nigel Adkins, sacked by Southampton last month, had been installed as the early favourite to replace Grayson, but Huddersfield maintained Robins had been their first choice and he was unveiled as the club's fifth manager in as many years at their Canalside training headquarters.
"It was a considered decision," said Robins, who alluded to the current rent dispute between the club, Arena Coventry Ltd - the owners of the Ricoh Arena - and Coventry City Council.
"I spoke to the owner, I sought assurances and didn't get any assurances from the owner of Coventry," he said.
"I absolutely wish them all the best, they work their socks off and as I've said before that club needs support from Coventry City Council.
"That led to a decision to want to speak with Huddersfield and that's how the decision has been arrived at."