The Cook Islands club last featured in the main draw back in 2001 and this is the first time they've progressed to the Group Stage via the Qualifier.
Sean Latimer put Tupapa in control with two goals in the opening 22 minutes and, along with Maro Bonsu-Maro, ended the match with a hat-trick.
Soakai Vea won a penalty for Veitongo after the half hour but lost his cool and was shown a red card.
Vai Lutu netted from the spot to reduce the deficit to 2-1 but the Tongan side were unable create further momentum with a man down, while captain Sione Uhatahi was also sent off late in the game as Veitongo finished with only nine players.
After an improved performance and dominance across the park, Tupapa coach Alex Napa was pleased his side secured such a resounding victory.
''Two wins is good to have in the bag, one more game to go and we hope that we can put a decent team in for the last game and make a good job of it at the last hurdle,'' he said.
''Sometimes heat has a way of leveling things a bit. I think for both teams it was just a bit frustrating, the heat got to both teams and I'm really glad that my boys composed themselves a little bit more.
"We could see it, it does that to players on a hot day like this and it's good to come away with a good result.''
Veitongo coach Timote Moleni said one positive for his side was how they played through the last 16 minutes despite having just nine players on the field.
''The boys played so well, they still lack of the ability to finish in front of goal which really impacted the match. We worked on their finishing in the past two days, I am sure, they are not confident enough.''
Meanwhile Lupe Ole Soaga got their campaign back on track with a 13-1 demolition of American Samoa's Pago Youth.
Striker Lapalapa Toni took just nine minutes to put his side on the scoresheet while Suivai Ataga was the star performer with five goals.
Coach Paul Ualesi believed the win was well-deserved after a well-structured performance.
''First of all I'm so proud of my boys for the hard work, they tried to execute our game plan well for this second game and it was good for them to work on their discipline as well and you can see we tried to press on them the importance of working on their discipline and controlling their mental aspect of the game as well," he said.
"But yeah, very happy to get a really good win."
Pago Youth coach Li'atama Amisone Junior admitted the team's weaknesses proved too big to overcome on the day.
"We did not keep our composure and stick to the game plan," he said.
"Everyone makes mistakes and has to learn from them. Indeed we've identified our weaknesses and fitness is the number one thing."
Lupe Ole Soaga seal the other remaining qualifying beth with a draw or better in their final group match against Veitongo FC.