Wolves lost 2-0 at Brighton and Hove Albion to finish second from bottom of the Championship, twelve months after they dropped out of the Premier League.
They became the only club in the English game to have twice suffered relegations from the top division to the third tier in successive seasons.
"Today is a sad day at the end of what has been an horrendous season," Morgan told the Wolves website (www.wolves.co.uk).
"As chairman of the club I take my responsibilities very seriously; we have failed our supporters, and the city, and for that I am truly sorry. Clearly a lot of mistakes have been made over the past two seasons.
"Although every decision has been with the club's best interests at heart, we have clearly got some of those decisions wrong. I understand the fans' pain, anger and resentment towards me, the board and the players, especially within the context of successive relegations."
Morgan, a Liverpool-born businessman who bought Wolves from Jack Hayward in 2007, has been criticised for his managerial changes in the past 15 months.
He sacked Mick McCarthy in February 2012 with Wolves mired in a season-long battle against relegation from the Premier League. McCarthy had kept Wolves in the top division for two seasons and, in finishing 15th in 2010, helped the club achieve their highest finish since 1980.
McCarthy was replaced until the end of last season by his assistant, Terry Connor, and after relegation the board appointed Norwegian Stale Solbakken.
Solbakken lasted just six months before the club turned to Dean Saunders, the former Liverpool and Wales striker who left Doncaster Rovers to take charge at Molineux.
Morgan made no mention of Saunders's future in his statement on Saturday, but he did point the finger at his players.
"We are all looking at ourselves in the mirror and reflecting on what went wrong," said Morgan, who was ranked 171st in this year's Sunday Times Rich List with a personal fortunate estimated at 510 million pounds.
"It is only right the players do the same. Many of them know they have not performed to the standards they are capable of and they must take their share of the responsibility.
"Next season is going to be very difficult," he added.
Saunders told the website: "As soon as that final whistle went I am now starting to rebuild the football club because we have to move forward now.
"I shook all the players' hands and from now on I am starting again. Sometimes it takes hitting rock bottom before you realise it has got to change, and I have got to change everything and rebuild the club."
- Sports & Recreation