Rins ran in the lead group from the off in the restarted 20-lap Austrian GP and was lapping faster than the Ducatis ahead of him, passing Pramac's Jack Miller for second on lap 10.
Going after Dovizioso, Rins went for a pass on the inside of the factory Ducati rider at Turn 6 on the following lap but lost the front of his Suzuki just as he took the lead. Prior to the race, eventual winner Dovizioso had touted the Suzukis to be a big threat on Sunday, with Rins admitting the feeling he had during the race was one he has "not felt many times" in his career.
"Everything was going very well, I honestly felt like winning the race," said Rins. "This feeling, I have not felt it many times, but I really felt [able] to win. It was a shame, but the two Suzukis were there, Joan [Mir] finished on the podium, a great result for the brand. So we'll see what happens this weekend".
Rins had a moment on the front end of his Suzuki at the same corner he crashed the lap before, and thinks his fall was a result of braking with slightly more lean angle than in previous tours.
"Perhaps the mistake was trying to stop the bike with more [lean] angle," Rins added. "From what we have seen in the telemetry, when I turned on the throttle I was much more inclined than in the previous lap, and that was what made me lose the front end."
Mir admitted pre-weekend he considered a rostrum charge "difficult", and thinks improved qualifying form and his start "made the difference" in his charge to a maiden podium in second.
"Well, before the weekend if you said I would be on the podium, I'd say it will be difficult for sure," said Mir. "But the truth is that bike works really well here, in FP1 I felt so good here in Austria. When you start with a good base it's easier to approach a little bit the weekend.
"It was a hard one, fighting with these guys [Dovizioso and Miller] is difficult. In sector one and two they are really strong, but then looks like our bike is quite competitive and quite good. So, a good podium for me. I think we made the difference in this weekend because we improved the qualifying and then the start was good.
"[That] was our weak point. Normally we have a good race pace, but we struggle [usually in qualifying] and it is difficult then to recover positions."