England defender Maguire's drilled effort in stoppage time added insult to injury for Saints, who had Pierre-Emile Hojbjerg sent off for two bookings.
“It's a fantastic feeling at the end to win away in extra time and to see all this character and mentality in the team," said Foxes boss Claude Puel.
“It's fantastic and I hope we can continue this moment. I said to the players this team (Southampton) plays very well and they deserved another result against Everton. They have a new mentality and more strength in the team.
“We know their quality. It was important for us to resist and defend very well in the first half. It was difficult to find space and movement but we had three or four good chances.
“After the second half, we found more space and after the goal Southampton played deep and allowed us to start with the ball to make some good play. We had a lot of chances to win this game."