The Hoops manager watched on as his side were beaten by the Austrians in their final group game earlier.
Going into the match, Celtic knew at least a draw would see them through to the knockout phase.
But they were unable to get that, with Salzburg deservedly taking a 2-0 lead late into the game.
At that time, Leipzig were beating Rosenborg in the other group game – meaning they would advance alongside group winners Salzburg.
However, Tore Reginiussen popped up with a late equaliser for the Norwegians.
And that 1-1 draw in Germany was later confirmed to put Celtic into the last 32.
Rodgers was obviously frustrated at a disappointing performance from his team, but he still insisted that Celtic deserved to go through when speaking post-match.
Asked by BT Sport if he was happy to go through, Rodgers responded: “Absolutely. There’s always that initial mixed emotion when you lose a game.
“But this campaign was over the course of the six games. You see how tough the group is, how good a team Red Bull Salzburg are and obviously you’re playing against a team like Leipzig from one of the top nations in Europe.
“And actually Rosenborg, to go there and get a result, and having beaten them twice, it’s been an absolutely brilliant effort by the players.
“To finish on nine points in a group like that, which was very tough, was an incredible effort through all the games that the players have been playing.
“So I’m so, so proud of them and obviously we deserve to get there because we finished second on the nine points.”
Celtic will now go into the last 32 alongside Salzburg – and they’ll find out who they get in the next round when the draw is made on Monday at 5pm.