Neser, competing in his first test in three years, broke the 96-run partnership between Ben Stokes and Will Jacks, dismissing both in his consecutive overs.
The right-arm pacer finished with five for 42 as Australia bowled out the opposition for 241 with a lead of 64 runs, setting themselves an easy target chased down in just 10 overs.
"I am sure I'll cherish this one," the 35-year-old told host broadcaster Fox Sports after his spell.
"It's a moment I'll never forget, having my family there.
"I think my sister's out there. I haven't seen her for a year and a half, so I am going to say hi to her."
Neser praised the support he received from captain Steve Smith, who plucked a one-handed catch in the slips to dismiss Jacks, and keeper Alex Carey, who stood up to the stumps to grab an outside edge from Stokes.
"This achievement would not have been possible without Carey and what Steve did there at the end," Neser said.
"I didn't even have to ask Carey to come up. He just does it. To have a keeper like that, it's great."
This is Neser's third game of the longest format, his last one being against the West Indies in Adelaide in December 2022.
He was included in place of Nathan Lyon for the pink-ball test, with regular captain and fellow fast bowler Pat Cummins still ruled out due to injury.
"I love the Gabba and the fans here. This is my home," he said. "Playing the Ashes is such a dream for Australia, and to be a part of that is special."
With the win, Australia take a 2-0 lead in the five-match series. The third test will begin on December 17 in Adelaide.