Despite absorbing most of the pressure and showing little attacking threat, a resolute performance from Gordon Strachan's men saw them stretch their unbeaten record to five games as Brown scored the only goal of the game with 13 minutes left.
The Celtic man had retained the captaincy as Darren Fletcher made his long-awaited return following illness but only to the bench, and he provided the decisive contribution on Wednesday.
Leeds United's Ross McCormack spearheaded the attack with Steven Fletcher for Scotland, while Norwich's Russell Martin returned to his unaccustomed role in the centre of defence.
A patient but spirited opening saw both sides play some neat football, with Scotland's David Marshall forced into the first save of the match after 11 minutes when he tipped a curling left-footed effort from Werder Bremen midfielder Ludovic Obraniak over the crossbar from 25 yards.
A midfield clash between Brown and Poland's Slawomir Peszko shortly before half-time prompted a head-to-head confrontation between the pair, with the referee taking a lenient approach in only booking both players.
That was almost the last incident of note in an open first half as Poland were kept at arm's length, but Scotland offered their first attempt on goal in injury time when a neat passing move ended with right back Alan Hutton hitting low at Wojciech Szczesny's legs.
With time running out Poland pushed for an equaliser and Marshall was forced to punch a long-range effort away, but Strachan's men held on.
