A last-four clash with Jose Mourinho's Roma awaits Leicester as its UEFA Europa Conference League campaign gathers steam, with Friday's (AEST) 2-1 win at PSV highlighting its class.
Maddison scored a 77th-minute equaliser in Eindhoven, after Eran Zahavi's opener in the 27th minute for the Eredivisie hosts.
Ricardo Pereira then hit the winning goal in the 88th minute after Patson Daka had a shot blocked, carrying Brendan Rodgers's team into uncharted territory.
The first leg between the teams had finished goalless.
"I'm so proud of the boys, so proud of everyone here," Maddison said.
"These nights, there's such a big build-up, and you get that big-game feeling.
"We started a bit slow probably, and we actually played a lot better once they scored. We settled a little bit more when we were one goal behind, which is not ideal but is how it went.
"In the second half I thought we were brilliant, so intense, playing on the front foot, penning them in, and we scored two goals of quality to win the game, and I think we were deserved winners.
"I'm really happy and really proud of the lads."
Maddison's goal was his third of the season in the competition, the most by a Leicester player.
"I got the goal at a perfect time," the England international said. "It was brilliant from Ayoze Perez to chop the guy and lay it on a plate for me.
"From then I felt it, like we were going to go on and win the game, wherever it was going to come from. It felt we were fitter, more intense, and we go there in the end with Ricky at the back post."