It was a typically scrappy affair between the two teams in the first half as they cancelled one another out in the middle of the pitch and not a single shot on target was produced in the first 45 minutes.
However, Brady's second-half strike sparked a revival for Norwich, who have claimed their first win in 11 games and in turn ended the Baggies' unbeaten run.
The midfielder took the resulting free kick, but he blazed it over the bar, failing to trouble Ben Foster, who has conceded four goals in his last three league matches.
Salomon Rondon has been in goalscoring form of late, but he struggled to make much of an impact in the first half as his early attempt from 12 yards out drilled wide.
The England Under-21 international got past the defender before taking a shot towards the top left corner, but his attempt flew wide of the post.
The best chance of the half came when Claudio Yacob had the ball laid off to him by Berahino after Norwich's defence cleared a corner. The Argentine unleashed a strike towards the top right corner, but he just missed the target.
Brady summed up Norwich's half by trying an ambitious 35-yard shot from the right wing, which blazed over the bar and caused some of his teammates to have a word.
At the other end, Stephane Sessegnon had a decent chance to score before the half-time interval after surging into the box, but he took too many touches before taking a shot, and by that time, Bassong had closed him down and forced the West Brom man to fire his shot wide.
Five minutes into the second half, Norwich changed the game when Brady put the Canaries 1-0 ahead thanks to some help from Matt Jarvis.
Tony Pulis's side failed to clear a ball into the box, which allowed Jarvis to latch on before sending the ball to Brady, who smashed it into the left corner of the net.
From that point, the game opened up more, but Norwich's attacking sense was revived and they looked comfortable on the ball, with little threat to their goal.
Patrick Bamford came close to doubling Norwich's lead when Gary O'Neil's whipped ball into the box found Brady, who then knocked it to Bamford, who turned sharply and unleashed a shot, but it slammed into the side-netting.
There were some concerns for Brady shortly after the half-hour mark when he was brought down by Dawson and fell awkwardly as a result.
Brady's neck was checked over and a stretcher came onto the field, but he opted to play on for a few minutes before leaving the pitch after complaining about his vision.
West Brom tried to find that all-important equaliser in the closing few minutes of the game, but they did not produce a single shot on target and Norwich were able to hold on to see out the win.
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