The goalless affair was brought to life with ten minutes remaining when substitute Vardy headed home Cengiz Under's cross.
It was Vardy's 11th goal in 12 Premier League games against Arsenal.
The Gunners amassed 12 shots and had the majority of possession but were unable to breach Kasper Schmeichel's goals.
Leicester now move up to fourth on the table, while Arsenal are in tenth.
Speaking after the game, Vardy said the win was something Leicester could build upon.
"We weathered the storm, we started getting into it, we played well and came away with a 1-0 win. Its big for us but its just the next game. We had a couple of bad results and just wanted to put it right.
"It was a good build up, James has seen me me free and he put it on a plate for me really.
"Luckily I made an impact and we take the three points home, it gives us a little boost, because the last two games were not good enough and now we can build on this."