Australia's Steph Catley (29), who stepped in as captain for injured Matildas striker Sam Kerr (29), scored a penalty in the 52nd minute to give Australia the lead.
Yet the 10th-ranked Matildas were forced to hang on for the win as they absorbed wave after wave of pressure from the Irish, who went close several times in the dying seconds at Stadium Australia.
"We are a team that's battling, Irish DNA is battling, that is within us," Pauw said. "I'm really proud for their first time on this stage...
"Next game, it will be clear that like this game we went in for results, to play for results, not just to be part of a nice tournament in Australia," Pauw added.
Marissa Sheva (26), who gave away the penalty, was substituted 10 minutes after the goal and was inconsolable for the rest of the night, wiping away tears while team mates wrapped her in hugs.
"She's okay," Pauw said. "I think if you play for the first time in the World Cup... and you get a penalty against, it's quite normal you're upset."
Pauw pointed out the narrow loss was another solid performance from the 'Girls in Green', who take on Canada on July 26th then Nigeria on July 31st in their remaining Group B games.
"We have shown that we can play against the very best teams in the world, we've now shown how to get through it," the Dutch coach said. "We do not fear anyone."