Things got complicated for the home side when Edinson Cavani pulled one back for PSG in the 62nd minute, leaving Enrique's men needing three goals to address the aggregate deficit as well as the away goal.
"This is a insane, unique sport. Children, and adults, here will never forget what happened here", he beamed during his post-match conference. "We didn't play in the first half", Al-Khelaifi said.
Barca have won their last six league games and scored 17 goals in their last three in all competitions, although Deportivo have improved under new coach Pepe Mel and should provide a sterner test for the Catalans than their last visit to Riazor, where they won 8-0 last season.
Barca manager Luis Enrique's gesticulated wildly on the touchline while nearly 100,000 fanatical supporters inside the Nou Camp roared their side on, but their hopes seemed lost, even when Brazilian playmaker Neymar drilled home a rocket-fuelled free kick on 88 minutes.
But what followed at Camp Nou on Wednesday will go down in European football folklore. "It was a horror movie, not a drama, with a Camp Nou that I have seen very few times as a player or coach". But instead of Messi, it fell to Neymar to conjure up the three goals now needed. "I'm convinced we'll be close to qualifying", he said.
FC Barcelona players celebrate after defeating Paris Saint-Germain in a UEFAChampions League last 16 clash.
The visitors held on until the 88th minute but Barcelona simply refused to be denied and when Sergi Roberto prodded home the side's third goal in a devastating seven-minute spell, Paris St Germain were left licking their wounds once more.
He said in Marca: "Messi will renew, I'm sure of that".
"We shouldn't be too comfortable because there were other teams that have lost 5-0 in Camp Nou".
Luis Enrique rattled the words off, trying to make sense of the game he had just watched, his pace quickening until eventually he stopped, sat back, puffed his cheeks out and said: "And that's it, here we are".
"I'm in a very good moment".
In addition, Barcelona, being down 4-0 from the first leg, were 10-1 to advance on Wednesday.
Barca defender Gerard Pique, meanwhile, predicted a surge in the birth rate in Catalonia, as occurred in 2009 following Andres Iniesta's stoppage-time leveller against Chelsea which took Barca into the Champions League final.