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.
For Poland, as expected, Borussia Dortmund striker Robert Lewandowski was missing and his club colleague Lukasz Piszczek took on the captaincy.
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.
Poland continued to force the pace and in the 28th minute another long-range effort from Arkadiusz Milik was pushed wide low to his left by Marshall.
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.
Strachan made two changes for the Scots at the break, bringing Darren Fletcher on for his international return, and Steven Naismith also replaced Steven Fletcher up front.
It was Poland who continued to press however, and they should have taken the lead in the 58th minute when a free-kick on the left from Obraniak was met by the unmarked Kamil Glik, who headed downwards but wide with the goal at his mercy.
A neat flick from Obraniak then released Milik again in the 65th minute, but he lifted the ball over Marshall and the crossbar from the edge of the six-yard box.
Against the run of play Scotland took the lead in the 77th minute with a brilliant goal from Brown.
A cross was only partially cleared by the Polish defence and it fell to the 28-year-old, who emphatically rifled the ball passed the stationary Szczesny from 25 yards.
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.
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