De Minaur looked down and out numerous times in his clash with Zverev, losing the opening set 6-2 before trailing 5-4 on the German's serve in the second.
However, he won that service game and went on to steal the set, and then saved a match point on his way to victory in the deciding 10-point tiebreak.
The win put Team World 5-3, and Rafter was particularly pleased with the win given it came over one of Team Europe's top players, who he felt had come out to send a message.
"Bloody oath it is," he said when asked by Flashscore whether the win was a particularly big boost as it came against a player as accomplished as Zverev.
"I mean, the way (Zverev) came out, he obviously showed he was the number one player or becoming the number one player in the world.
"He wanted to stick it to us - 'how dare you beat us last year' - and it's just not going to happen. He came out like that, and he saw the fight of Alex.
Team Europe headed into this year's Laver Cup as the favourites thanks to the presence of Zverev and Carlos Alcaraz, the two most decorated players and only two Grand Slam playing the tournament.
However, Rafter feels that Team World's fighting spirit gives them a real chance.
"(Team Europe) have got Alcaraz as well, so you're playing the two best players in the world right now, and we've got our guys just going to bust their arse to get a result," he added.
"It goes a long way. Never underestimate how important that is."
Rafter and De Minaur are the two Australians on the team, and the vice-captain shared his pride in De Minaur playing "the Aussie way".
The two-time US champion, who has been critical of Nick Kyrgios over the year, emphasised the importance of his nation's players committing to it, and the dangers of those who don't.
"I wouldn't call us the prettiest players, but we sort of found a way to get the job done and to fight no matter what," he said.
"We've had a few disruptors amongst our team at times, which I haven't been a real big fan of, because keeping the type of culture that's built up over many years - it will get destroyed very quickly.
"To see this kid come along - he's been there for about 10 years, playing the Davis Cup. We need that. Going back to grassroots, it's a great example for young kids coming through.
"We've seen it here, in the Davis Cup. Just his attitude, it's a great role model."
Team-event specialist
De Minaur also beat Zverev at last year's Laver Cup, and has claimed a number of important wins for Australia in team events.
He said he hugely enjoys being part of a team, and that it brings out the best in him.
"I just thrive in these types of environments," he said after the match.
"I look forward to them every single year. The tennis tour is a long year, and whenever you get to play for a little bit more than just yourself, it brings that kind of extra energy.
"I've always played really well representing my country, and I'm glad that I was able to put up a good performance today as well."