The Foxes, who have now won all four away games this season, grabbed the lead in the second minute when Vardy intercepted a poor back-pass from Leeds defender Robin Koch and set up Harvey Barnes to slot home.
Brendan Rodgers' side doubled their lead in the 21st minute when Marc Albrighton whipped in a cross and Vardy's low header struck Leeds keeper Illan Meslier in the face and fell to Tielemans to fire home.
Leeds came out strongly after the break and pulled a goal back three minutes after the restart when a deep cross from Stuart Dallas went through a crowded box and into the far corner of the net.
Pablo Hernandez then hit the bar for Leeds with a fine effort from the edge of the box but Leicester restored their two goal advantage with a beautifully worked third goal.
Substitute James Maddison found Cengiz Under who showed great finesse to pick out Vardy who slotted home.
Maddison was involved with the fourth goal, in stoppage time, when Mateusz Klich challenged him from behind and after a VAR review, referee Andre Marriner awarded a penalty which Tielemans confidently converted.
"I thought my players were brilliant. We had to be tactically disciplined and then play when we had the ball," Rodgers said. "If we get to March, April and we are in that position [second], then it will be nice, but there is a long way to go."
Leicester are on 15 points, just one point behind leaders Liverpool.
Leeds are 12th on 10 points and their Argentine manager Marcelo Bielsa struck a self-critical tone.
"In the first 30 minutes we defended poorly and after we didn't create enough danger. That is one of the main responsibilities of the manager," Bielsa said.
"I am not taking the blame, just stating the players were not correctly distributed. The players are disappointed."