Brodka, skating in the 17th pair, crossed the line in 1 minute 45.00 seconds, setting a benchmark that looked virtually unbeatable with just three more pairs to race.
However, the Dutchman, who went in the 20th and final pair, had other ideas, straining every sinew in the last few laps, and when he crossed the line the clock showed a dead heat.
After about 20 seconds of nervous waiting it was confirmed that Brodka had won by just 0.003sec in 1:45.006, denying the Dutch yet another success at Sochi's Adler Arena.
Brodka - the first ever Polish speed skater to win Olympic gold -- raised his arms in triumph: “I can't believe this. The world championships and the Olympic Games -- those were my dreams, and I now I've got the gold medal,” said the 29-year-old Pole.
“I was nervous before the race. After the finish I didn't know it right away. When I saw I had won, it was an unbelievable feeling.”
Meanwhile, silver medallist Verweij, 23, struggled to contain his disappointment at missing out on gold: “It wasn't what I wanted, it wasn't enough. I should be happy winning silver but this feels like a huge loss,” said the Dutchman.
“At the moment it feels like all my hard work and my consistent races were for nothing,” he added. “After such a season you know you can do it and I felt pretty good.”
The USA’s Shani Davis, a two-time Olympic champion over 1,000m, could only finish 11th.
The Americans had entered the speed skating competition in Sochi with high hopes of winning several medals, but have yet to make the podium.
By contrast, Verweij’s silver means that Dutch skaters have claimed 13 of 21 medals handed out in the first seven days of competition in Sochi.