The Potters took the lead on the half-hour mark when Adam drilled in from Nzonzi's cut-back.
Six minutes later, referee Kevin Friend deemed a sliding tackle from Brown on Adam that won the defender the ball - and replays suggested any contact between the players had been minimal - to be worthy of a straight red card, a call that left Sunderland boss Gus Poyet fuming.
The Uruguayan saw his team battle bravely thereafter, but Stoke's win was sealed in the 81st minute when Nzonzi slotted home having been set up nicely by Peter Crouch.
Mark Hughes' side, who had been 17th at the start of the day and separated from the relegation zone by goal difference alone, moved up to 14th after this result.
Meanwhile Sunderland, who are three points adrift of safety, now occupy bottom spot in the table with Crystal Palace - watched from the stands by their new manager, ex-Stoke boss Tony Pulis, as they won 1-0 at Hull - having moved above them.
Jack Colback saw a shot deflect behind for a corner as Sunderland exerted some pressure early on, but it amounted to nothing and at the other end, Adam sent a free-kick on to the roof of the net.
Moments later Geoff Cameron delivered two dangerous crosses into the Black Cats box in quick succession, one of which came to Marko Arnautovic, whose strike on the turn was blocked.
Steven Fletcher then brought the ball down - seemingly with his arm - in the Potters area and, with no whistle blown, produced a shot which Asmir Begovic got out to parry.
Soon after, Stoke skipper Ryan Shawcross had to clear off his line as a hoisted ball forward ricocheted towards goal and Arnautovic then appealed for a penalty having fired an effort against John O'Shea, but none was given.
Stoke continued to press and were rewarded with a goal when Nzonzi latched onto Cameron's chip down the right, turned Ki Sung-yeung and laid the ball to Adam, who was arriving in the box.
The Scotland midfielder side-footed a low shot that gave Vito Mannone little chance of keeping it out.
It got worse for Sunderland as Brown was given his marching orders by Friend, whose decision prompted an angry reaction from the defender's team-mates and manager.
Brown, sliding at a considerable speed when he challenged Adam and won the ball, appeared to make little contact - if any - with the Stoke man.
Sunderland's complaints went unheeded and their opponents looked to swiftly make the most of the man advantage, but could not add another goal before half-time.
The hosts continued their search for a second after the break but Sunderland managed to keep them at bay and brought a save out of Begovic via Phil Bardsley's attempt.
Stoke then broke and Adam advanced towards goal, but he blazed his shot well off-target from outside the area.
Sunderland did their best to battle back and, after Adam Johnson shot wide, Fletcher went to ground in the box when the boot of Begovic, coming out to clear, connected with his chest, but again no spot-kick was awarded.
Nzonzi then wrapped up the three points for Stoke, finishing first time from a through-ball by Crouch, who advanced forward having collected Adam's pass.
And the Potters might have had more goals had substitute Andy Wilkinson or Arnautovic been able to convert late chances.