Brown's 77nd-minute goal delivered a third consecutive away win for Scotland, deciding a tight friendly in Warsaw. Strachan dismissed a suggestion it would have any significance when the countries meet again in the Euro 2016 Group D campaign in October, but he said the 1-0 win was further evidence that Scotland have become hard to beat.
"I've no idea how far we have come in a year," he said. "I don't think you can quantify that. But we are hard to beat and I want to take it on a stage further. The players have to take that next step, it's on their shoulders.
"It's up to them, and the ones maybe not here tonight, to take us a stage further so that when we have the ball we're a real threat. At the moment we're trying to retain the ball better so the other team has less chances because we have more possession.
"I thank the players for putting so much effort into it, it's easy at this time of year just to slacken off. We were a bit fortunate, we understand that, but there were a lot of things we worked on. We try to be more potent when we get the ball back, we need to be more confident, more trusting when passing the ball into people like we did for the goal, where we were brave and passed it to people and made defenders defend.
"We need to do more of that. But we are beginning to understand it a bit better. Teams find it hard to play against us and that's great, that's a start. What we have to improve on is the attacking side. I'm sure the next time we meet there will be more attackers on there from both sides.
"Will it have any bearing on the next Polish game? Absolutely not. We know that. It could have gone either way tonight. But what we did learn was that we saw the stadium and the support they are going to have, and the enthusiasm of the team we played against. That's a benefit for us. To get acclimatised to one or two things here."
Darren Fletcher returned to the Scotland team as a second-half substitute, his first cap in 16 months after dealing with ulcerative colitis. Andrew Robertson, Dundee United's 19-year-old team-mate, also came off the bench for his debut. For Strachan, though, it was Brown who was the architect of the night.
"He's ungainly in a way but he does what all good midfielders do: pass the ball, keep it well, move it on. But he also has what a lot of midfielders don't have, the power to dribble and the power to win the ball. He can be really pleased with himself." I thought the midfield all did well."