Having slipped to a 1-0 defeat in the first leg, Craig Shakespeare's side put in a spirited display at the King Power Stadium in the return fixture, only to lose out 2-1 on aggregate.
And England international Drinkwater admitted his frustration on missing out on a place in the semi-finals, with the players starting to believe they could even reach the final in Cardiff and lift the trophy.
"We did believe we could win it. We weren’t in just to take part," he told reporters.
"We’re a squad that believes we can do things some people probably didn’t think we’re capable of, but we just keep pushing ourselves on the training field and out on the pitch.
"Now we’ve had a taste of it, we want more. There’s nothing for us to be scared of."
Leicester captain Wes Morgan echoed the thoughts of his team-mate in regard to the squad being able to return to European competition in the coming years.
The Jamaica star returned from a back injury on Wednesday, and he believes the club still has the players to qualify for the competition again next season.
And Morgan feels the Foxes's star players will not turn their back on the squad after the end of their incredible journey.
"It has been an incredible journey. I’m not sure who thought we could get this far, but we believed in ourselves and we achieved something massive.
"To get all the way through to the quarter-finals in our Champions League debut season is a massive achievement.
"This season has been more downs than ups, but I feel we’ve turned it around and we’re coming good now. If we had shown the same form all season that we’ve shown of late, we could be in a Champions League spot again.
"First of all we need to concentrate on the rest of the season, hopefully finishing on a high note, and then next season we need to dust ourselves off and look forward again.
"I know everyone loves it here and we’re a family here, so I’m sure everyone wants to stay."