Lyttle was keen to conclude the Irish League season over the coming three weeks but he said the Cliftonville board decided otherwise.
"Cliftonville didn't see that as appropriate," said Lyttle, 39.
"That was their decision and I totally respect that."
After next week's final regulation Premiership game against new leaders Linfield, the Reds will be involved in the European play-offs which could mean two further matches.
Lyttle said that Sligo were also prepared to let him finish the Irish League season with the Reds.
"I didn't want to walk away and leave it as it was," Lyttle told BBC Radio Foyle.
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"But there's a board there [at Cliftonville] and they decided the right thing to do was to freshen it up and look at Mal [Donaghy] and George McMullan.
"I would have been happy to work through the next couple of games and try to reach the target I set out at the start of the season which was European qualification."
Lyttle said that the lure of a full-time managerial role convinced him to apply for the Sligo role left vacant following Dave Robertson's dismissal earlier this month.
Former Linfield and Newport County manager Warren Feeney was also believed to have been interviewed for the Sligo job along with a couple of cross-channel candidates.
"I was just a wee Irish League part-time manager so to come up against some big names and get the job was an honour and a privilege," added Lyttle.
The former Reds boss watched Sligo draw 1-1 with Derry City at the Showgrounds on Saturday although caretaker Declan McIntyre remained in charge for a final time.
Gerard Lyttle has said Cliftonville rejected his request to remain on as Reds boss until the end of this season following his appointment as Sligo Rovers manager.