The Reds have made a promising start to the campaign, winning their first three games, including victory over Manchester United, and were top of the Barclays Premier League table coming into this weekend's round of fixtures.
Their next test comes at Swansea on Monday, where Rodgers will face his former employers at the Liberty Stadium.
Some were sceptical when Rodgers was appointed as Kenny Dalglish's successor in May 2012, but Laudrup believes the Northern Irishman is showing signs of restoring the Anfield club to the top four, claiming a Champions League spot for the first time since the 2009-10 season.
"It is hard to say if they will finish in the top four," said the Dane. "But they have started well, winning the first three games and beating Manchester United.
"We will still have to see how things will turn out as it is early in the season, but they are very close to Arsenal and Tottenham.
"They are definitely a better side than last year, the players that came in last year have more experience, and they have added some quality in Philippe Coutinho and Daniel Sturridge, who have played a lot of games and settled in.
"They are a lot closer to Arsenal and Tottenham than they have been for a couple of years and they will be battling for the top four."
Another man to swap south Wales for Merseyside last summer was Joe Allen.
The Wales midfielder made a strong start to life at Anfield following his big-money move, but a mixture of a loss of form and injuries have prevented him making the sort of impact he would have wanted.
Laudrup has no doubt Allen has the talent to succeed with the Reds, despite only working with him very briefly at Swansea, but has warned it is his mental toughness which will ultimately prove key.
"There are a lot of things to consider," he said. "Joe had only played one year at the highest level and then went there for a huge amount of money for a young midfielder.
"It was a lot of money and money brings pressure. The media and the fans have expectations because of the price, but you have to accept it.
"You cannot just play well once in a while like you can at the smaller clubs. That's the difference. He played a lot in the beginning but they have brought players in and there is a lot of competition but he has got a lot of talent.
"But I can only judge what I see on the pitch, I only had him for four days here after he came back from the Olympics so I do not know him well enough to have a big picture of his personality.
"But a big part of it is the mental side, because we have seen so many players make that step and find they cannot do it and then they go back down the ladder."