French striker Odsonne Edouard put the visitors ahead in the Europa League Group B tie after less than two minutes.
The Austrian champions, who reached the semi-finals last season, dominated thereafter and two goals in six second-half minutes from Munas Dabbur and fellow attacker Takumi Minamino, both from cut-backs, turned the tie around.
Celtic winger James Forrest, pictured, was sent off by referee Serhiy Boiko in the 72nd minute for denying skipper Andreas Ulmer an obvious goalscoring opportunity and Dabbur slotted home the consequent penalty.
Rodgers said: “I think the best team won. In the first half we started well and by the end of the half we had frustrated Salzburg, you could see in their game and play and we carried a threat when we broke out of our shape. So the first half I was reasonably pleased with, the second half we gave away disappointing goals.
“That was the frustration for us and then from that they gain confidence. And, when it gets to 3-1, it is a difficult game for us with ten men.
While Salzburg top the group with six points, Celtic have three from their win over Rosenborg at Parkhead.
The Scottish champions now face a double-header against RB Leipzig. The German side beat Rosenborg 3-1 in Norway last night.
Rodgers added: “You can see how Salzburg play, they are a very good side but we limited them in terms of clear chances. But at this level you have to be better with the ball and keep the ball and be able to play quicker with minimal touches, otherwise you don’t get a rest in the game and that hurt us this evening.”