The Foxes had looked set to be frustrated as they missed several chances in the first half while Wilfred Ndidi struck the bar early in the second period.
But Vardy finally broke the deadlock as he finished off an excellent team move, firing in his 11th goal of the season after Harvey Barnes and Youri Tielemans cut open the Arsenal defence.
Maddison then made sure there was no way back for Arsenal with a low strike through Héctor Bellerin’s legs and into the back of the net.
Victory means Leicester move up to second with 26 points from 12 games, five behind leaders Liverpool.
Arsenal, meanwhile, are now eight points off the top four and are without a win in the Premier League since beating Bournemouth 1-0 on 6 October.
Eighteen months ago, former Arsenal chief executive Ivan Gazidis hailed the arrival of Emery as a coach who “plays an exciting, progressive style of football that fits Arsenal perfectly”.
That description was certainly accurate about one of the teams on the pitch at King Power Stadium on Saturday, but it was not Emery’s side.
The Foxes possess a clear identity and a togetherness that appears to be absent from this current Gunners side who, apart from occasional flashes, rarely troubled their opponents.
Leicester dominated possession from the outset and even though they missed a number of decent chances – Ayoze Pérez firing just over after capitalising on a defensive error and Maddison putting a free-kick on the roof of the net – there was always the sense that the Foxes would find a way through.
