Rossi, a nine-times world champion, was third in the rain and wind of the Qatari desert after starting from 10th on the grid on a weekend when Vinales, 16 years his junior, dominated on his first outing for Yamaha.
"It wasn't an easy start to the championship but I was very happy to be able to get on the podium," said Rossi. "Now we go to Argentina. This is a track that I like very much and I'm happy to go there. I hope that there are good asphalt conditions because last year it was difficult for us."
Rossi was second at the hot and dusty Termas de Rio Hondo circuit in Argentina in 2016, having won 12 months earlier. But his former team-mate Jorge Lorenzo, whose decision to move to Ducati opened the door for Vinales at Yamaha, believes that the flying Spaniard could end up making Rossi feel his age this season.
"It's not easy for Valentino to keep racing MotoGP at his age," Lorenzo said. "The second oldest rider is maybe 32, and he's 38. I guess when you become 38 every season is tougher than the one before; that's something Rossi recognises himself. You don't see many 38-year-olds as fast as Rossi.
"It's not something easy and even less when you have an outstanding and strong teammate like Maverick."
However, former Suzuki ridfer Vinales, who became the second youngest rider to win in the premier class for two different manufacturers, just behind Mike Hailwood, recognises that he is already a marked man.
"We have the difficult challenge to continue getting good results, Qatar was only the first race of 18," he said. "I can feel the pressure a little bit because the level of my rivals is very high, but that motivates me even more to keep working hard. We have to be cautious, work step by step and race by race."
The Argentina race is also a key test for defending world champion Marc Marquez after the Spaniard managed just fourth place in Qatar on his Honda.
"I must fight for the podium," said the Spaniard, a double winner in Argentina and world champion in 2013, 2014 and 2016.