A Mitch Duke header in the first half at the Al Janoub Stadium, where Tunisian fans vastly outnumbered the Australian contingent, was enough to seal three points for Arnold's side, who defended well under intense pressure.
The victory also marked the first time since 1974 that Australia had kept a clean sheet at the World Cup, which Arnold described as "crazy".
"For the fans back at home, it's a moment they will remember for the rest of their lives. I just want them to enjoy. I said to the boys before the game, 'Let's put a smile on the nation's face'," Arnold told reporters.
"One or two teams that bring the nation together are the Socceroos and the Matildas. When the Socceroos play at World Cups, AFL fans, rugby league fans, cricket fans - they all become football fans.
"I can imagine the celebrations at home. Especially with the prime time TV 9 o'clock kickoffs (and playing) until 11, I think there will be a few hangovers in the morning. I won't have one!"
Australia have not reached the knockout stages since 2006, and things looked bleak after they were thrashed 4-1 by reigning champions France in their group opener.
"I do believe that the best 'friendly match' we could have ever had was against France because we got punished for our mistakes," Arnold said.
"Today, obviously the opposition wasn't as good as the world champions but I felt that our performance, our fight, our grit and our determination - the old Aussie way - was very important and I'm very, very proud of the boys."
Victory puts Australia's qualification for the knockout stages in their own hands.
"We've got to see what the result is tonight with France against Denmark," he added. "Nothing will change for us, we will focus on ourselves.
"I got them in a circle to tell them I'm very proud with what we've achieved - the nation must be very proud - but we've achieved nothing (yet).
"I don't want the players sitting up all night looking at social media, I need them sleeping and recovering well to get in the right mindset for Denmark."
As for the former players who had criticised Arnold and the team, he was succinct with his response.
"They've never coached. I haven't seen who was critical but I think some of them have never even been to a World Cup," he said. "So they don't have any effect on my life."
