Reports in some Sunday newspapers had suggested that Dalglish would leave the manager's position during the summer, with Wigan boss Roberto Martinez being tipped as the man to replace him.
After watching Danny Graham's 100th career goal condemn them to an eighth-placed finish, Dalglish was asked if he expected to be in the Anfield hot seat next season.
He replied: "I expect the owners to have more dignity and integrity than to believe a story in a newspaper."
The visitors were outplayed during the first half but posed a greater threat after the break, with Swansea goalkeeper Michel Vorm making one stunning save to keep out Andy Carroll's acrobatic overhead kick and then making a low stop to deny the £35million striker.
And Dalglish felt his side had deserved at least a point, but acknowledged their campaign, which ended with Liverpool below local rivals Everton for the first time since 2005, had not been good enough.
"I think on the second-half performance, the least they deserved was a point," he said.
"We've got to keep believing and in credit to the players they have kept believing.
"The passing and moving at times has been excellent and the goalkeeper's save to stop Andy was fantastic. We just have to keep going and maybe next year the luck will change for us.
"We won't run away from the points tally or hide or say the points tally is satisfactory to anyone who works for or supports Liverpool Football Club. We will face up to reality. The performances have been excellent in a lot of cases but not the points tally, so we will try our best to correct that."
The Reds were without captain Steven Gerrard after he failed to sufficiently recover from a back problem.
But Dalglish is confident the midfielder will be fit to be part of England manager Roy Hodgson's Euro 2012 squad.
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He said: "It's not my concern, but I wouldn't think he'll be a problem."
For Swansea the result capped a fantastic debut season in the Barclays Premier League, which has also seen them claim memorable wins over Arsenal and champions Manchester City. They missed out on a top-10 finish to West Brom on goals scored.
Unsurprisingly there was a celebratory atmosphere throughout the contest, with hordes of fans taking up manager Brendan Rodgers' invitation to dress up in Elvis Presley costumes.
And the Northern Irishman admitted the win, secured by Graham's sweet finish from Angel Rangel's pass four minutes from time, was the perfect way to cap the campaign.
"It has been a fairytale and what an end to the season," he said.
"The players did brilliantly, their pressing and work-rate was phenomenal.
"Everyone associates us with our passing, movement and creativity, but behind it there is a real hunger to get the ball back and the hunger was incredible today [Sunday].
"We were worthy winners and it finished a brilliant season. We have kept 14 clean sheets, which is outstanding at this level and to finish having beaten Arsenal, Manchester City and Liverpool in our first season gives us a big boost going forward."
The victory marked the final Swansea appearances for loan stars Steven Caulker and Gyfli Sigurdsson as they return to Tottenham and Hoffenheim respectively.
But Rodgers is hopeful of bringing the pair back to south Wales next term, having reportedly had a bid to make Sigurdsson's move permanent rejected.
"The season has finished and it's about planning going forward.
"In football terms the squad will break up as players will leave and players will come in.
"I will speak to Tottenham on Steven Caulker, he has been incredible for me this season at just 19 years of age.
"With Gylfi, we will have talks with Hoffenheim over the course of the next week or so.
"That's the challenge for me, recycling the squad, retaining that hunger and giving the players help."