The Scotland midfielder's only pre-season action was an hour in the Hoops' last match against Union Berlin last Friday night after recovering from a groin injury but he declared himself ready for the trip to north Belfast.
Brown turned in an impressive performance as goals from Mikael Lustig, Georgios Samaras and James Forrest ensured there will be little chance of an upset when the part-time Irish side travel to Parkhead next week for the second leg.
"That's my first 90 minutes since the Scottish Cup final and that's the main thing," said the former Hibernian player. "I am definitely going to feel the benefits of that. I feel fine. The more matches I get the fitter and more confident I will become."
Brown was pleased with the way the Scottish champions went about their business, especially on a synthetic pitch.
He added: "It was always going to be hard for the lads playing on an artificial surface. We trained on it for a couple of days just before coming over here but at the end of the day, the result was great for us. We played really well and got the three goals."
Cliftonville boss Tommy Breslin conceded the tie was over but he is looking for his side to make a decent fist of it in the return game in Glasgow.
He said: "It was always going to be an important part of preparation for our league season which starts in a month's time.
"We will lick our wounds, have another week's training under our belt, try to put on a credible performance and not try to concede too many.
"I thought the three goals were preventable and at this level you get punished. But they are a class act and that's the bottom line."
- Sports & Recreation
- Georgios Samaras
- Mikael Lustig