Ederson could do little in the City goal as Vardy and Tielemans fired three penalties past him. Vardy and Maddison added two from open play while goals from Riyad Mahrez and Nathan Ake goals did little for City's hopes.
City had not conceded five goals in a home game since February 2003 against Arsenal and it was the first time their boss Pep Guardiola had seen a side he is managing concede five in what was his 686th game in the dugout.
The hosts came out of the blocks the much better team at the Etihad Stadium out of the blocks the much better team, opening the scoring through former Leicester City player Mahrez on four minutes.
A perfectly weighted corner from Kevin De Bruyne was headed out of the box by Leicester but only as far as Mahrez who, unmarked, waited for the ball to bounce before curling it beautifully into the top right corner with his left foot.
City went on to enjoy over 80% of the possession in the opening 25 minutes but failed to capitalise on that time with a goal.
Rodri appeared to have scored after 35 minutes when he got his head to a De Bruyne free kick but replays showed he was offside and the goal was disallowed.
It was a frantic minute as Leicester immediately went on the attack. Harvey Barnes ran the ball down the pitch and put a good ball into Vardy who went down in the box under pressure from Kyle Walker. Michael Oliver pointed to the spot and the 2019-20 Golden Boot winner converted.
Vardy added to his tally nine minutes into the second half when Timothy Castagne put a low cross into the box and he got in front of his man to chip the ball over Ederson.
Eric Garcia conceded a second City penalty on 57 minutes when he brought Vardy down in the box. The 33-year-old stepped up once again and while Ederson went the right way, there was no way he could stop Vardy's fast and low shot going into the bottom right corner.
Maddison added Leicester's fourth on 77 minutes after receiving the ball in space on the left. He cut inside and curled the ball past Ederson with his right foot.
Ake managed to pull one back for City on 84 minutes after he headed the ball into the net from a set piece but it wasn't to be enough as Tielemans converted a penalty moments later.
"We have set up to infuriate them and it has worked a treat. We knew if we stuck to our plan we would get some chances," Vardy said after the game.
"It's always cat and mouse. It's a gamble both ways. I know where I am going to put it and he knows where he is diving. [I'm] over the moon to get the goals.
"It's only a start. The way we finished last season was not good enough so we have to keep it going now."
Despite looking uncomfortable in his movement in the latter stages of the match, Vardy confirmed that it was a minor bone issue that was niggling at him and not something more serious.
The defeat snapped City's 11-match unbeaten home run in the league dating back to last season, while it was also their first defeat in an opening top-flight home game since a 2-1 loss to Southampton in 1989.
Information from Reuters contributed to this report.