Competing at their first-ever major tournament, Georgia were deemed rank outsiders to progress to the knockout rounds after being drawn in Group F alongside Portugal, Turkey and Czech Republic.
Despite losing their opening group fixture to Turkey, Georgia picked up a point in a 1-1 draw with Czech Republic and needed to beat Portugal - who had already sealed top spot - to boost their hopes of reaching the last 16.
Portugal made wholesale changes to their starting lineup and Georgia ran out as comfortable winners against the Euro 2016 champions in Gelsenkirchen, with Kvaratskhelia opening the scoring after just 93 seconds with a clinical left-footed strike across the face of goal.
Georgia were then awarded a penalty for a foul on Luka Lochoshvili, following a VAR review, and Georges Mikautadze stepped up to net his third goal of the tournament and ultimately seal a historic victory for Willy Sagnol's side, who advance to the last 16 as one of the four best third-placed teams.
Kvaratskhelia was named Man of the Match following his standout performance for Georgia and the Napoli winger believes that his country's supporters played a key role in their memorable triumph over Portugal.
Kvaratskhelia: 'Georgia fans make us strong'
"The fans have given us so much support," Kvaratskhelia told UEFA.com after the match. "So many of them have been in Germany and that's one of the factors that contributed to our victory.
"I want to thank them all, including those who are supporting us from Georgia. We've done this together and unity makes us strong, that's what we've shown today."
Reacting to receiving his Man of the Match award, the winger added: "There was no individual best player today, it was the team, and we proved to everybody that we can really play.
"This is the best day in the lives of Georgian football fans. We've made history - no one would have believed we'd make it happen.
"No one would have believed we could beat Portugal, but that's why we're a strong team. If there's even a one per cent chance, we've shown we can make it happen."
Gerogia's reward for finishing third in Group F is a challenging last-16 tie with three-time European Championship winners Spain at the RheinEnergieStadion in Cologne on Sunday.