The Foxes reached the quarter-finals as they overcame Brighton in a penalty shoot-out on an emotional night at the King Power Stadium.
Wednesday’s tie fell on the third anniversary of a helicopter crash at the stadium, which killed Khun Vichai and four others.
A crowd display and minute’s silence took place before kick-off ahead of the first home match to fall on the date, which finished 2-2 after normal time before Leicester won 4-2 on penalties.
“I am pleased we could win on the third anniversary of such a tragedy that is for everyone around the world, but none more so for people here in Leicester,” said Foxes boss Rodgers.
“Time has gone very quickly. We had a lunch yesterday with the staff and players, and people told stories of what he was like.
“He has given the club this status around the world and Khun Vichai made what is the modern-day Leicester City Football Club.
“We had a moment reflecting in the memorial garden before the match on our responsibility to continue his legacy. I never met him sadly but it was one of the reasons I came in, in a funny way. I’ve felt as if I have known him.
“The club and supporters have done brilliant tonight and I am just really pleased for them because it has been a really poignant day.”
Harvey Barnes gave Leicester an early lead before Adam Webster equalised in first-half stoppage time. However, there was still time for Ademola Lookman to restore the hosts’ advantage before the half-time whistle.
Substitute Enock Mwepu deservedly headed Brighton level midway through the second half to set up the drama of penalties.
Neal Maupay crashed his spot-kick against the crossbar before Mwepu saw his effort saved by Danny Ward, with the Foxes scoring all four of their kicks through James Maddison Barnes, Patson Daka and Ricardo Pereira.
Rodgers, who made 10 changes from the team that won at Brentford in the Premier League at the weekend, said: “I think the players were great as a lot of them had not played a lot of minutes over the weeks.
“We showed moments of quality and scored two really good goals. We also responded well to conceding the equaliser and that showed spirit, and then we had the poise at the end with the penalties. The penalties were really good and we showed really good composure.”
Brighton boss Graham Potter revealed the injury Dan Burn suffered late in the first half at the King Power Stadium is a “concerning” one.
The big defender fell heavily after an aerial challenge and did not come out for the second half, with Mwepu taking his place.
“We are a little concerned to be honest. We will have to get it scanned tomorrow (Thursday),” said Potter.
“We won’t know until then but at the moment it is a little concerning because of the way he landed from quite a height as well. So we will have to wait and see.
“It’s another disappointment as I felt we did enough in the 90 minutes to win the game, that was my feeling. So I am a little disappointed not to win it in normal time and very disappointed to go out.
“When you make changes like we did you sometimes don’t get that cohesion and fluidity but I felt we did tonight and the players who came in did well.”