The 30-year-old scored to ensure the Foxes still have a chance of reaching the quarter-finals following a 2-1 first-leg defeat at Sevilla on Wednesday.
Claudio Ranieri's side were outplayed for the majority of the game but the hosts only had goals from Pablo Sarabia and Joaquin Correa to show for their domination.
Kasper Schmeichel saved Correa's early penalty and kept Sevilla from running riot with several stops before Vardy converted Danny Drinkwater's cross in the 73rd minute.
Vardy and wife Rebekah had Finley last month and after scoring his first goal for 748 minutes, the forward sucked his thumb in celebration.
"Everyone worked hard and gave everything, the celebration was for Finley, he's probably asleep but that was for him," he told BT Sport.
Defender Danny Simpson also praised Vardy's intervention and believes the Foxes showed shades of last season's title-winning form.
"I'm happy for Vards, he deserved his goal, he worked his socks off. It was a great move, a great cross and that's where he wants the ball put – and he'll score," Simpson said.
"At 2-1 we've given ourselves a chance. We looked like a threat again on the counter.
"It was us from last season. Going 2-0 down at a place like this, we kept going because we knew how important the away goal might be."
Sevilla come to Leicester on March 14 for the second leg of the last-16 tie and Simpson feels the Foxes showed character after their problems in the Premier League, where they sit just a point above the relegation zone.
"It's over two legs so for the first 90 minutes of 180 away from home it was a tough place but I think that away goal was really important," Simpson told LCFC TV.
"The way we performed and worked for Leicester and the fans, with the way the season has been going, was massive for us."
