The Falcons scored five tries in their best display of the season to end a five-match losing run.
"I'm really pleased with the result and the way the boys played," said Bates.
"We had a really good first half and turning round 14-13 up in those conditions was really excellent.
"We pretty much dominated in most phases of the game. Our defence was outstanding, we scrummaged really well and our lineout functioned well.
"We got the ball forward much more clinically and if we had held on to the ball a bit more the score could have been really something.
"I was disappointed that sometimes we try a little bit too hard and we need a bit more patience in some areas of the game, but when we get it right and towards the end when they were quickly we showed what we were capable of."
Bates is looking forward to their last-four clash even though some of his players struggled through the tie against Castres.
"We do have a couple injuries and hope to sort them out before the semi and we go to Worcester with a lot of confidence," he added.
"Jamie Noon had a bit of recurrence of his ankle injury and he fell awkwardly when he nearly scored but we will look after him and make sure he is okay.
"I thought the pack were outstanding tonight and Tom Dillon came on for Noon and showed what he can do. Carl Hayman showed some great touches in the match and he scrummaged so well that we had a great platform
"We will bring some different guys in for next week against London Irish. Some of the guys need a rest and we want to have a look at some other combinations.
"The result means our season is still very much alive but we do have a lot to work on for next season but there is a huge amount to play for and I'm very excited by the potential. Tonight (Saturday) was a step in the right direction but we can be far more clinical."