Substitute Andy Carroll's strike for the hosts set up a tense finale but Leicester were worthy winners.
Maddison blasted Leicester in front in the 55th minute, and when the outstanding Tielemans produced a great finish to double the lead after 72 minutes it seemed that Leicester's seventh away win in the league was in the bag.
Carroll's first Premier League goal for Newcastle for 10 years offered a late twist.
"When we got the goal against us, we started panicking and stopped playing really. In general, we played well and the win is the most important thing," Tielemans said.
Leicester began well, and Maddison came close to giving them an early lead with a curling effort that fizzed just wide.
Manager Brendan Rodgers was livid when Jamie Vardy and Tielemans were guilty of over-elaborating once they had sliced open Newcastle shortly after, while Vardy also had a goal from a tight angle disallowed for offside.
Newcastle were well-organised without offering much in attack in a largely forgettable first half, but the hosts started the second half with greater purpose.
They were stopped in their tracks though when Harvey Barnes surged forward and slipped a pass to Vardy who held the ball up before teeing it up for Maddison to lash home.
Leicester's slick attacking football was evident again for their second goal as Marc Albrighton slid an astute pass inside towards Tielemans, who did not even need a touch before burying the ball past a helpless Karl Darlow.
Newcastle manager Steve Bruce sent on Carroll, and the big striker rewarded him almost immediately with a clinical volley -- his first league goal since returning to the club in 2019.
Suddenly the hosts came alive and Leicester were living dangerously at times until they were relieved to hear the whistle.
Newcastle's winless run in the league now extends to five games and they stay in 15th spot with 19 points.
"All game it was there and obviously when I came on, we got the goal and put the pressure on them so could have nicked something. It wasn't a great first half. It's hard this time of the year with all the games," Carroll said.
"We came out in the second half and played a lot better. I thought we deserved something."