Sunderland's fate looked set when Bournemouth's Josh King side-footed in the winner at the Stadium of Light in the 88th minute after a swift counter-attack.
A Hull defeat would have given Sunderland a stay of execution and when Southampton were awarded a penalty at St. Mary's in the final minute there was a glimmer of hope for David Moyes's team.
But Hull keeper Eldin Jakupovic dived low to save Dusan Tadic's penalty in the final minute and earn the 17th-placed Tigers a vital point in their own battle against the drop.
Sunderland had some chances but top scorer Jermain Defoe was not at his sharpest and, not for the first time this season, the home supporters flooded out of the ground before the final whistle.
"My feeling at the start of the season was it was going to be a hard graft," Moyes told BBC Radio.
"But ultimately today I feel for the supporters. They pay their hard-earned cash to come to the stadium and we have not given them enough this season," he said.
Moyes said he was not ready to discuss his future, saying two or three weeks of reflection were needed for the club before a plan was developed for a return to the Premier League.
The club's American owner Ellis Short issued a statement saying such a plan would be developed once the season was over and admitting the club had made mistakes.
"I acknowledge that during my ownership mistakes have been made, particularly in the area of player recruitment, and as a result we have found ourselves struggling to survive in recent seasons.
"We had massive disruption during the summer transfer window and an unprecedented number of injuries throughout the season. These are difficulties which we have been unable to overcome and we are paying the price for that now," said Short.
“We need to improve, both on and off the field, and despite the bitter disappointment of today there is a strong determination to do so throughout the club. There is significant work to be done over the summer and when the season is concluded, we intend to share our plans to move forward with our supporters," he added.
Middlesbrough, who play on Sunday at home to Manchester City, are in 19th place on 27 points while Swansea, who are at Manchester United on Sunday, are on 31 points and Hull are on 34.
At the Hawthorns Jamie Vardy struck the 43rd-minute winner for champions Leicester against West Brom in a 1-0 win.
Vardy finished with confidence after Shinji Okazaki latched on to a poor back pass from Salomon Rondon.
Goals from Ashley Barnes and substitute Andre Gray helped Burnley to get their first away win of the season with a 2-0 victory over Crystal Palace. The result moves Burnley to eight points above the relegation zone and should secure their top-flight status for another season.
Burnley move up to 14th place, above Palace and West Ham, who drew 0-0 at Stoke City.
On Sunday, leaders Chelsea are away to Everton while second-placed Tottenham are at home to North London rivals Arsenal.
(Editing by Clare Fallon)
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