The Red Devils were staring down the barrel at another disappointing exit until late goals from record scorer Romelu Lukaku and captain Youri Tielemans forced extra time.
After 30 more minutes which Garcia compared to "two boxers (who) just kept fighting and fighting", Tielemans lashed home a penalty in the 125th minute to win in dramatic fashion.
Garcia even described the victory as a "remontada", the name given to Barcelona's famous comeback against Paris St Germain in the 2017 Champions League.
"Football is emotions, this evening we had plenty," Garcia told reporters. "When you're down 2-0 in the 83rd minute, it's never easy to come back and win the match.
"But this is what I said to the players at the hydration break: we had to score the third goal in the match and then anything is possible.
"We did it, we delivered and then it opened up the game again."
He added: "It is a scenario that can make the group grow even stronger and tighter together and realise that up until the match is not finished, until the final whistle, a lot of things can happen as we just demonstrated today."
Belgium had long laboured under the weight of expectation, but their 'golden generation' could never quite deliver the trophy their abundance of riches promised.
Some of them remain – including Lukaku, playmaker Kevin De Bruyne and goalkeeper Thibaut Courtois – though Garcia said his team did not boast quite the same depth. But he said this Belgium side are still a force to be reckoned with.
"I arrived 18 months ago because I believe that there's quality in this team. It's not the best of all time, but tonight we wrote history," he said.
"We didn't win anything, we qualified for the round of 16 and we'll see who we're going to face ... It doesn't matter who it's going to be. Now we're going to savour our victory and our 'remontada'."