Claudio Ranieri's side had gone two games without a win, but they are off the mark now thanks to a 2-1 victory over Swansea City at the King Power Stadium.
Vardy struck to score a sixth goal in his last seven top-flight matches, while captain Wes Morgan doubled the team's lead in the second half.
The hosts showed attacking intent from the first whistle, but they were not helped along by the rainy conditions which eventually produced isolated puddles on the pitch.
Robert Huth came close to opening the scoring in the second minute when the defender directed a header towards the near post, but goalkeeper Lukasz Fabianski comfortably snatched the ball with his gloves.
As expected, Vardy was heavily involved in the Foxes' attacks, making runs in behind, trying to catch out Swansea's defenders, but despite Leicester's dominance, the visitors managed to hold them off for the first half-hour.
Mahrez was given an opportunity from a free kick on the left side of the final third, but his effort ended up curling over the crossbar.
It took 25 minutes for Francesco Guidolin's men to cause a threat at the other end of the pitch as Gylfi Sigurdsson attempted to swerve the ball past Kasper Schmeichel, but the goalkeeper made a relatively easy stop.
There was a case for a penalty just before the opening goal when Huth was held in the box, but referee Roger East opted against blowing his whistle, despite new rules stating that grappling in the area can result in a spot kick.
The breakthrough came in the 31st minute when Vardy, who had not registered a single shot on target in his first two games, found the back of the net with a clinical finish.
The England international, who was the league's second-highest scorer with 24 last season, darted away from Jordi Amat before slotting home.
Swansea tried to respond instantly through Sigurdsson, who unleashed a curling shot, but it was not on target as the ball drifted over the bar.
The visitors had a short spell of dominance as they attempted to pepper Leicester's goals with shots, but their final touch was lacking and Leicester almost doubled their lead before the half-time whistle.
Five minutes after the break, the Foxes nabbed a second goal when Morgan lashed in a volley from six yards out after Swansea failed to clear a corner.
Not only did the captain give his team a two-goal advantage, the goal also marked his 200th appearance for the club.
Leicester continued to dominate and they were handed an opportunity to score a third when Jordi Amat gave away a penalty by bringing down Okazaki in the box.
Mahrez, who netted from the spot in his team's opening league match against Hull City, stepped up to take it, but Fabianski went the right way and denied the Algerian before Okazaki fluffed his rebound shot.
There was an injury concern for Leicester when Schmeichel was forced off the pitch after colliding with Fernando Llorente, who spent most of the match isolated up front.
Ron-Robert Zieler came on to replace the Dane, but he was rarely used as Leicester continued trying to hunt down another goal.
Vardy - full of confidence - produced a shot from distance while going for glory, but the ball kept rising as it edged closer to the net and ended up going into the stands.
Despite dominating the game, Leicester and their fans had a nervy final 10 minutes after Leroy Fer pulled a goal back for Swansea by heading home from Modou Barrow's inswinging cross.
They were able to hold on, though, to register their first win of the campaign, while Swansea will return to Wales having recorded back-to-back defeats in the league.