Matchwinner Dempsey told Sky Sports 1: "We had more space, more time on ball and we were able to open them up.
"We've not been able to score enough goals but we got one tonight and it's an important win."
The goal came when the United States international followed in after visiting keeper Jose Reina uncharacteristically spilled Danny Murphy's low shot.
And Dempsey said: "You've always got to take a chance. Danny did a good job getting the shot off and you've got to be in the right place at the right time. Sometimes you get lucky and it falls to you.
"Any points right now are big because the table's so tight. We're just happy to get the win."