Rossi took the lead when Marc Marquez retired with a comfortable margin just before half-distance.
The seven-time MotoGP champion Rossi then seemed well positioned for a first race win since Assen in June 2017.
But Rins managed to stay close to Rossi’s tail and with four laps to go he whipped the inside of his Yamaha opponent in the fast left turn at Turn 7 to take the lead.
Rossi tried to fight back but ran too far at Turn 12. He was never able to get close enough to Rins again to make one more attempt to regain the advantage, and eventually settled for second place.
“I’m very sorry for [losing] the victory, “said Rossi in Parc Ferme.” When I saw Marquez fall, I said, ‘Maybe I can do it’, but then Rins came along.
“To be honest, I got the maximum out of it, the race was very fast, but in the end he was better than me, he drove very well and I couldn’t overtake.
“It’s a shame because I haven’t won in a long time, but we’re still strong and we’ll try next time.”
Not only did Rins take his first MotoGP win, but also ended a Suzuki victory drought that dates back to Maverick Vinales’ success at Silverstone in 2016.
“I’m so happy I have no words to explain these feelings,” said Rins emotionally.
“Thanks to Suzuki they did a really good job [making] a good bike all these years.
“I gave my maximum, I’m so happy, the feeling is incredible.
“I beat Valentino, that’s incredible for me, he was my idol when I was very young.
“When we did the pre-season preparation, I certainly believed that we could win a race. We have a strong bike. Victory was very close in Qatar, in Argentina we started from behind.
“In the warm-up it was a disaster for us. We tried a different setup and it was worse and we went back to the standard setup for the race and it was amazing.”