Australia-born Tuipulotu was instrumental in the Scots' 27-13 win over the Wallabies, shaking off an attempted Suaalii tackle for the hosts' first try.
Rugby league convert Suaalii later got his man with a bone-crunching tackle near the Wallabies try-line but injured his wrist in the contact and had to come off the field at the half-hour mark.
Before he exited, Tuipulotu fired a few choice words at the young midfielder, who shot back a few of his own, including: "See you next time."
Asked about the prospect of renewing hostilities in the series opener in Brisbane on Saturday, Suaalii played the rivalry down.
"(It was) just footy. Once you do things on the footy field, you keep them on the footy field," he told reporters in Brisbane on Monday.
"I'm ready to go, ready to play. Not too much to say about that.
"I always focus on myself first before I put my energy into someone else.
"Obviously, there are fans who want to see a rivalry and all that but, at the end of the day, it's a team sport."
Suaalii will be closely watched in the series, not least because of the reported A$5 million deal he signed to defect from the 13-man game.
The tall and athletic 21-year-old impressed in his first tests on the Wallabies' season-ending tour of the Northern Hemisphere and is already seen as a marketing success for Rugby Australia.
Many home fans remember rugby league convert Israel Folau's famous Wallabies debut in the first test of the 2013 Lions tour, when he scored two tries in a losing cause at Lang Park, and went on to score 37 tries in 73 appearances.
They will hope Suaalii can have a similar impact in his first Lions outing at the same venue on Saturday.
"He's someone I admire for the person he is and the way he plays the game," Suaalii said. "He did kill it in that 2013 series.
"It was a dream of mine to play the Lions because of watching him, Kurtley Beale, all these different players do their thing in the game. And it kind of sparked something in myself."
Suaalii was joined by big lock Will Skelton at the Wallabies' press conference but drew most of the media's attention.
Skelton, who missed the Wallabies' last-gasp win over Fiji with a calf strain, confirmed he was fit and available to play at Lang Park.