Evans, 53, was relieved of his duties at Elland Road on Tuesday.
The Scot was the sixth head coach to work under Cellino since April 2014.
"In the last week or two it became obvious that I was not going to be around. I don't feel disrespected, I had a good working relationship with Mr Cellino," Evans told BBC Radio Leeds.
"The club secretary Stuart Hayton conveyed to me that my contract would not be extended beyond the end of the season and it was handled in a very professional phone call.
"I spoke to Massimo earlier in the summer but not in recent days or weeks.
"I certainly felt we had done enough to go forward but you don't make that decision as a manager. That lies with the owner and the board. I have to accept it, as much as I am saddened by it."
MK Dons boss Karl Robinson and Bristol Rovers manager Darrell Clarke have both turned down the opportunity to succeed Evans.
The former Rotherham boss, who had promised fans he would secure a play-off finish next season, believes the club need to hire an experienced British coach.
"It has to be a strong character and someone who is going to be very focused and understanding of what the remit is," he said.
"I was very clear that if Leeds United's aims for next season were to finish in the play-offs then that would be happening under Steve Evans and Paul Raynor.
"I can only take that the benchmark has been raised after the statement that Massimo put out yesterday, so if I was a Leeds United supporter I would be getting excited for a strong British coach."
Former Leeds head coach Steve Evans says he was content with the way his departure was handled despite not being told by chairman Massimo Cellino.