"Firstly, I'd like to apologise to everybody, the supporters, for coming up short and ultimately not having what was needed," said Coleman, according to Chronicle Live.
"It was a brutal experience, the disappointment is huge. Maybe it was blind optimism with three games to go, but even with five minutes to go [today] we were in the driving seat, we were winning.
"We didn't play very well today. The five games previously, we were very, very good, but today you could see both teams were nervous and edgy.
"We were in the lead, the atmosphere in the ground was great, everyone was singing because it looked OK, and then it all changed. We are 44 games into the season and we haven't been good enough.
"Sorry to the supporters who have stuck with us and come here every week and followed us away from home, we haven't been good enough – we've fallen short, and I apologise to them."
Sunderland's relegation will now see them participating in the third tier of English football for only the second time in the Black Cats' history.