Celtic's Scott Brown was the Scotland goal hero for the second game running, as he followed up his match winner against Norway in November with another decisive strike to settle the friendly in Warsaw.
The game was arranged well in advance of the UEFA 2016 qualifying draw that pitted the two nations together in Group D, with Strachan's men due to return to Poland in October.
And while Strachan lauded his captain Brown, he believes it would be foolish to read anything into the result.
"It was not a great game and you can't expect anything other than that at this time of year," said Strachan.
"Scott Brown was the man of the match and the goal sealed it.
"But it meant absolutely nothing as these two sides will not be the same for the Euro 2016 campaign, but it gives us confidence and builds nicely to our next friendly."
Brown retained the armband, despite the return of Manchester United's Darren Fletcher, who made his first appearance for his country since October 2012 as a second-half substitute after being sidelined through illness.
But it was a captain's performance from Brown, who was dogged in midfield, although he was perhaps lucky to stay on the field after a first-half altercation with Slawomir Peszko, after which both players were booked.
When Brown's winner arrived in the 77th minute, it was a thunderous effort passed Arsenal goalkeeper Wojciech Szczesny.
"It was with my left-foot as well," said Brown.
"One of those that could end up in row Z, but fortunately it went in.
"We are doing what the manager is telling us and we are looking like a team out there.
"It is good to get a game against our group opponents and show what we can do against them."
