Jamie Vardy had a first-half goal ruled out for offside and volleyed wide from a narrow-angle soon after the break, before setting up the 55th-minute opener.
Receiving the ball on the left side of the box on the counter-attack, Vardy teed up Maddison, who powered a rising drive past goalkeeper Karl Darlow.
Leicester doubled their lead in the 72nd minute as Marc Albrighton cut the ball behind Newcastle’s defenders to Tielemans, who whipped a first-time finish inside the right post.
Newcastle substitute Andy Carroll gave Newcastle some hope, volleying in to score his first goal since returning to the Magpies in August 2019, but the hosts were unable to equalise.
Leicester have equalled a club record by winning five successive away Premier League matches against an opponent, and they climb to third, one point behind first-placed Liverpool and Manchester United, in second.