Adam Swift scored a hat-trick of tries, with further scores coming from Francis Meli, Sia Soliola, Jordan Turner, Tommy Makinson and Lance Hohaia in the 46-10 win.
James Roby - fit and looking sharp again after a recent ankle injury - was another impressive performer, making the tries for Soliola and Hohaia.
Six St Helens tries before half-time made the second 40 minutes a formality.
Brown said: "I thought the first half was really pleasing and we came here expecting a really tough game. They are a big side with crafty players.
"Roby was great and you can't beat hookers like him in that kind of form. It's a crucial spot and not to have him makes it a lot harder.
"Roby had a great start. Lance Hohaia did too. We were playing well on the right edge and we got it going. Everyone loves working off him (Roby). He played 65 minutes and that's only his second game back."
The clash with Leeds will take place on Friday.
"We have some confidence now," Brown said. "We have a big task against Leeds but we should go there thinking we can do okay.
"The guys want to do well and it will be a really good game. We want to stay alive and give ourselves a good shot at Old Trafford.
"Swifty is playing well, but so is Tommy (Makinson). He's got speed and he's come through our system too."
Hull KR head coach Craig Sandercock explained the absence of Cory Paterson, who now heads to Wests Tigers, was due to a burst appendix.
"Saints were way too good, too big, too fast and too classy. Saints were full strength and we were missing eight regulars," Sandercock said.
"We missed Cory Paterson who burst his appendix yesterday - what can you do?
"We didn't perform. We had too many errors. They were way too good for us.
"We made some very poor defensive mistakes but to the guys' credit the second half wasn't too bad but we didn't perform in the first half. The game was over by that point."