With 13 minutes remaining, the Iceland international received a pass from Kurt Zouma, beat James Maddison with a Cruyff turn and unleashed a magnificent, dipping 30-yard strike into the top-left corner – his fourth goal in as many top-flight games clinching a 2-1 away win.
Sigurdsson's 50th Premier League strike, which came after Wes Morgan had been sent off for the hosts, was an early candidate for goal of the season and sealed the Toffees' first away win under Marco Silva.
"Our first away win this season, two [league] victories on the bounce now and three very good points in what is a tough place to come to," said Sigurdsson, whose side also beat Fulham at home last weekend, before falling to Southampton on penalties in the Carabao Cup on Tuesday.
"They had 10 men and had a lot of players behind the ball. It was hard to break them down so as soon as I had the chance to score I thought, 'Why not?', and thankfully it hit the back of the net.
"It's definitely up there [as one of my best goals]. It came at a very important time and was a really good goal that I'm very happy with.
"I've said it for the last few weeks that we didn't feel like we were too far away from putting back-to-back results together. We felt like we were close to winning games and it's nice to win two [league] games on the bounce."
50 - Gylfi Sigurdsson has scored his 50th Premier League goal. Since his debut in the competition, no player has more from outside the box (19, level with Philippe Coutinho). Nifty. #LEIEVE— OptaJoe (@OptaJoe) October 6, 2018
Silva enjoyed a happier visit to the King Power Stadium than last season, when he was sacked as Watford boss after a 2-0 defeat in January.
Discussing Sigurdsson, who now has as many goals in eight league games this season as the four he managed in 27 last term, Silva said at his post-match media conference: "It was a fantastic moment for Gylfi.
"He showed how god he is in those moments, how he received the ball and shot was fantastic. I know his quality - if you can create and build a style of play that can fit players like him it will help. We deserved the three points."
Everton (@Everton) October 6, 2018
As well as the attacking midfielder, Brazilian winger Bernard shone on his first Premier League start, his sublime skill creating Richarlison's early opener for Everton.
Sigurdsson said: "He's a fantastic player. He's got creativity and agility and two good feet, good vision and quality as we've seen so far."
Leicester, meanwhile, have won four and lost four in an up-and-down start to the campaign, this defeat leaving them level with Everton on 12 points from eight games.
Manager Claude Puel said: "We are disappointed of course, we didn't start the game with enough tempo, it's another time we conceded a goal after six minutes. It's too difficult after and we are giving ourselves a mountain to climb, we have to correct this.
"Our second half was better, the beginning of the second half had good intensity, we had good chances and we had chances to score to make the difference and after the sending off, it changes the game. Both teams can win the game and the sending off changed the game and gave them the opportunity to win."