Both sides showed a real lack of quality in the final third as neither goalkeeper was forced into any difficult saves throughout the match.
And to make matters worse for the home side, Brown was given a straight red card by referee Martin Atkinson for a late challenge on Ricky van Wolfswinkel in stoppage time.
The result sees the Black Cats now three points adrift at the bottom of the table, with Norwich remaining in 14th.
The opening period was a cagey affair, with neither side finding any rhythm as the ball was given away far too often.
Gus Poyet's men were restricted to efforts from range, with Fabio Borini flashing a powerful strike just wide of John Ruddy's goal.
The visitors were also not afraid to shoot from distance, with Gary Hooper and Nathan Redmond both denied by smart saves from Vito Mannone.
Leroy Fer had the best chance of the half, side-footing a loose ball too close to Mannone from a good position just before the break.
The second half followed a similar pattern, with both sides struggling to find that killer pass to break the deadlock.
The introduction of Steven Fletcher did add impetus to the home attack, with the hitman dropping deep to try to create, as well as making more probing runs.
Brown brushed the outside of the post with a close-range header after a melee in the Norwich six-yard box, with Ruddy beaten, as the home side pushed for a winner.
Then two great chances in the space of a minute at both ends added some rare excitement to the encounter.
First, Fer again found himself in space, and from the angle forced Mannone into a smart save low to his left.
Then the home side broke clear, with Adam Johnson beating Ruddy to the ball near the corner flag, and with the goalkeeper stranded, Johnson picked out Ki Sung-Yueng, but the South Korean could only loft the ball onto the roof of the empty net.
Brown's sending-off late on summed up the frustration of the Black Cats, and with the busy festive period coming up, he will be a real loss to Poyet's survival bid.