Since Dalglish returned to Liverpool to replace ousted manager Roy Hodgson in January, the Reds have taken 28 points out of a possible 42 to climb up to sixth place and now have an eye on fifth and a place in the Europa League.
Saturday's 5-0 win over Birmingham was the 16th successive Premier League game they have scored in - the best run in the top flight - and Lucas feels the rejuvenated Reds can overhaul fifth-placed Tottenham.
"I think the way we are playing at the moment we deserve to be higher," Lucas said. "Tottenham still have an advantage because they have one game in hand and are three points ahead but we have to make sure we win every single game and if we do we will have a happy end to the season."
Liverpool's rise up the table has been an achieved at the same time as integrating some young players into the squad. In the absence of injured skipper Steven Gerrard, 22-year-old former youth team captain Jay Spearing has slotted in alongside Lucas and the Brazilian has been impressed by his new partner.
"We know it is very difficult when it is your first season in the Premier League but he has come in and played really well," Lucas said. "Always we are under pressure to win because we still think we can get fifth place but he has done well and is developing very quickly - and the same goes for the other younger players.
"If you see the younger players I am already old but I am still learning. When I arrived in Liverpool a lot of people helped me to learn and that is what I am doing now, trying to give Jay some confidence on the pitch.
"The team looks good and when fans see more young players coming in and playing well that means Liverpool have a bright future."
Liverpool play Tottenham at Anfield on May 15 in a match that could decide fifth place in the Premier League.