For the last few weeks the Reds have had to put up with their near-neighbours, who have finished higher only once in the Premier League era, occupying the place above them.
But the gap is only a point and if Dalglish's side better the Toffees' result at home to Newcastle they will snatch back seventh place on the last day.
"We've got a desire to go there and win and, as we've said before, we want to do as well as we possibly can in every game we play," said the Scot.
"We're not happy finishing below anybody in the table. We want to be finishing above everybody.
"Whoever is in our road, it is up to us to be better than they are and get there when we can - but it's not been too bad (a season) for us.
"To win a trophy (Carling Cup) after six years was a good result for us, to get to the FA Cup final was a good result, and to get into the Europa League is also a measure of success."
When Brendan Rodgers' took his side to Anfield earlier in the season they came away with a draw, thanks in part to the performance of goalkeeper Michel Vorm.
Dalglish is among the many Swans admirers this season, their first in the Barclays Premier League.
"They've been fantastic this year," he added.
"We had a good game against them earlier in the season at Anfield and they've done really well, kept to their beliefs and that's stood them in good stead.
"The good work was started off with Roberto Martinez and has carried on with the people who are there now.
"I think every manager who has come up has stuck to his own principles - you can only be what you are yourself, you cannot be somebody else.
"You've got a better chance if you just be yourself. I'm sure everybody that has come up has been like that - some have just been more successful than others, and Brendan has been successful.
"Their position in the league doesn't mislead anybody. They are a good side and it'll be a difficult game for us."