Brodka clocked one minute 45 seconds with Koen Verweij of Netherlands crossing the finish line in the same time in the final heat.
Brodka and Verweij waited nervously for a few moments while organisers went down to thousandths off a second to split the duo, with the Pole proving victorious in 1:45.006 compared to 1:45.009.
Brodka, 29, grabbed the Polish flag and took off on a lap of honour to celebrate his country's first speed skating medal of the Sochi Games.
Denny Morrison of Canada took bronze to go with the silver he won in the 1,000 metres on Wednesday.
There was no change in fortunes for the American skaters, though, despite a change of suits minutes before the start of Saturday's race.
The International Olympic Committee permitted the team to switch to an old World Cup suit and ditch their original Sochi hi-tech outfits.
Brian Hansen finished seventh, Shani Davis 11th, Joey Mantia 22nd and Jonathan Kuck 37th.
- Sports & Recreation
- Koen Verweij