Tonga trailed 28-0 but rallied with two fine tries from fullback Telusa Veainu, who looked sharp on his return from a nagging foot injury.
Argentina lost its opening game to France. So Mario Ledesma's lineup needed this bonus-point win ahead of next weekend's game against Pool C leader England, the 2003 champion which has already beaten Tonga and the United States.
This was a comfortable win, but the way Tonga got back into the game and pushed hard for a third try might concern Ledesma.
His team changes included dropping mainstay flyhalf Nicolas Sanchez and handing Benjamin Urdapilleta his first start since June 2013.
Montoya had scored a try as a replacement in the 23-21 loss to France, and took his tournament tally to four after replacing former captain Agustin Creevy in the starting XV.
These might prove the easiest tries he'll get, such was Tonga's poor defending early on. Santiago Carreras was also a new pick on the left wing, and the 21-year-old got his first international try in a one-sided first half where Urdapilleta also slotted four conversions.
"Just errors, poor skill, bit of miscommunication. Typical Tier Two errors," Tonga coach Toutai Kefu said. "You can't let Tier One teams get ahead like that. But I'm proud of how the boys fronted up physically."
A rout appeared imminent at this point with not even 30 minutes played but Tonga woke up with a well-taken try from Veainu — a neat side-step into the left corner from the Leicester Tigers favourite.
"He offers us an X-factor out wide," Kefu said. "It's great to have him back. Having a player of his caliber and class is fantastic for us."
Flanker Tomas Lavanini prevented a second Tonga try on the half-time buzzer, barging winger David Halaifonua out of bounds in the left corner. The try was ruled out, although there were questions over the tackle.
"I thought it was a shoulder charge. If we would have got that (try) we would have come out in the second half with a lot more belief and confidence," Kefu said. "There were a couple of 50-50 (decisions) today we didn't get."
After the break, the pace of the contest dropped at a humid 22,000-capacity Hanazono Rugby Stadium.
Argentine scrumhalf Tomas Cubelli had a try ruled out in the 52nd minute for a knock-on, following a referral to the television match official.
Tonga improved, and Veainu did well to squeeze into the right corner for a try confirmed after a review.
Sanchez went on as a replacement and missed a late penalty in a poor second half from Argentina.
Tonga faces three-time runner-up France next weekend.
Kefu sounded confident Tonga can repeat its upset World Cup win over France eight years ago.
"I think so," he said. But, "But we need to address those errors and be a lot more clinical to be competitive."
Argentina 28 (Julian Montoya 3, Santiago Carreras tries; Benjamin Urdapilleta 4 conversions)
Tonga 12 (Telusa Veainu 2 tries; Sonatane Takalua conversion).
HT: 28-7