The captain, who also netted the only goal in a November friendly in Norway, lashed a powerful first-time effort high into the net in the 78th minute as Scotland stretched their unbeaten record to five matches.
Darren Fletcher made his international comeback as a half-time substitute following a 16-month absence through a chronic bowel condition as Scotland largely frustrated their hosts, although Kamil Glik and Arkadiusz Milik both missed good chances to open the scoring.
Scotland rarely threatened, although Alan Hutton and Ross McCormack both had chances, but they were comfortable for the most part and the win will boost optimism over their chances of qualifying for the European Championship.
Strachan sprang several surprises in his team selection, notably employing a 4-4-2 starting formation with McCormack joining Steven Fletcher up front.
David Marshall was preferred to Allan McGregor in goal and Charlie Mulgrew lined up at left-back with Barry Bannan on the right of midfield and Ikechi Anya on the left.
The match started with the roof closed at Warsaw's National Stadium, where Scotland will return in October in a European Championship qualifier.
Marshall made two decent first-half saves from long-range shots but was otherwise untroubled.
His first stop came after an even first 10 minutes when he tipped over Ludovic Obraniak's curling effort after the creative forward had sidestepped James Morrison.
The Cardiff keeper later pushed wide a bouncing effort from Milik after the striker had been afforded too much space 28 yards out.
Scotland's only other first-half scare came when Poland right-back Lukasz Piszczek ran in to meet Gordon Greer's headed clearance but the Brighton defender charged out well to help force the skipper to blaze wide from 18 yards.
Scotland created their first chance in the closing seconds of the half when Brown committed a defender in the box and played Hutton in through the right channel. The right-back had three team-mates closing in on goal but could not find a pass and elected to shoot from a tight angle, but Szczesny blocked at his near post.
Darren Fletcher came on at half-time for Morrison while Steven Naismith replaced Steven Fletcher.
The visitors were soon under pressure from Obraniak's deep left-wing cross but Piszczek volleyed well over.
The game was opening up and Slawomir Peszko shot straight at Marshall before Mulgrew released Anya down the wing, but the Watford player crossed behind the well-placed Naismith.
Scotland soon carved a better chance when Bannan's pass allowed Naismith to run at the outnumbered home defence. The Everton player laid the ball off for McCormack 18 yards out but Piszczek threw himself in front of the shot and blocked.
Poland should have scored in the 58th minute after Hutton was booked for bringing down Peszko. Obraniak's free-kick handed Glik a free header eight yards out but the centre-back sent it a yard wide.
Scotland were cut open following a quick passing move as Obraniak lifted the ball over the Scotland defence for Milik to run on to but he shot over under pressure.
Dundee United teenager Andy Robertson came on for his debut midway through the half along with fellow full-back Phil Bardsley.
Strachan then switched to a 4-2-3-1 line-up by sending Charlie Adam on for McCormack and the substitute was quickly involved in the goal.
Naismith held up a throw-in before laying it off to Anya, whose right-wing cross found Adam. The Stoke player's header was half-cleared and Brown ran in to power a shot into the roof of the net from 18 yards.
Marshall made injury-time saves from Michal Maslowski and Lukasz Teodorczyk before the final whistle sparked jeers around the stadium.