Swansea came close to scoring in the first half when Jonjo Shelvey was dispossessed by Gylfi Sigurdsson, who shot narrowly wide.
Liverpool were more of a threat after the interval and Andy Carroll’s acrobatic bicycle kick was punched away by Swansea goalkeeper Michel Vorm before Graham finished from close-range in the 86th minute.
It was the Reds’ tenth defeat away from Anfield and left Dalglish’s side in eighth place, four points behind Merseyside rivals Everton.
“I think after the second-half performance, the players didn’t deserve what they got – the least they deserved was to come away with a draw,” Dalglish told Liverpool’s website.
“Even right at the death when there was a collision between their goalkeeper and two defenders, the ball still didn’t even drop for Luis Suárez.
“We’ve just got to keep – and the players have kept – believing what they’re doing to be correct. The passing and the moving has at times been excellent.
“The goalkeeper made a fantastic save from big Andy’s overhead kick. We’ve just got to keep going and maybe next year the luck will drop in for us.”
“It’s a great credit to the players that they do keep working and do keep believing. It’s the last game of the season and it would have been easy for one or two to get the Ambre Solaire out and have an easy afternoon, but they did not want to lose the game.”
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