The on-song striker scored an important brace for the Hatters which, added to goals from super-sub Harpal Singh and skipper Keith Briggs gave manager Chris Turner plenty to smile about.
Bramble gave Stockport the lead in the eighth minute with a superb bicycle kick, but Junior Mendes and Scott McGleish hit back to give Northampton the lead.
However, David Hunt was then sent off for a second bookable offence and the game swung into Stockport's favour.
Singh netted early in the second half to level matters, before Bramble and Briggs sealed the points with well-taken goals.
Stockport were given the perfect start after Bramble's moment of brilliance.
The striker was perfectly placed in the centre of the penalty area as Northampton defender Chris Doig made a mess of clearing his lines and headed straight to him. Bramble met the dropping ball with a superb bicycle kick and Northampton goalkeeper Lee Harper had no chance as the ball flew into the right-hander corner.
It could have been 2-0 a minute later, but Briggs headed over after a good cross from Gavin Strachan out on the right-wing.
However, a slip from young defender Michael Raynes in the 11th minute nearly gave Northampton an equaliser. Striker Mendes latched on to a pass on the edge of the box after Raynes fell, but his powerful right-footed shot hit the base of the right-hand post.
Bramble produced another bicycle kick in the 15th minute, but this time Harper made an easy save.
Northampton then got on top. Mendes turned Stockport loan defender Joe Dolan at the near post and somehow forced his shot under the body of Stockport keeper Carl Ikeme to level matters.
Strachan forced Harper to make an athletic save four minutes later after a sweet volley which looked destined for the top right corner.
But McGleish should have given Northampton the lead midway through the half, heading over from Eoin Jess' cross.
McGleish was denied by an excellent save from Ikeme a minute later having broke free on the left edge of the area.
The pressure finally told and McGleish got on the scoresheet, reward for all his hard work. He met a cross from Martin Smith at the back post, slid in and deflected the ball past Harper.
Stockport continued to battle and Briggs saw a shot saved after a pass from Greenwood before the interval, which Northampton went into with ten men after Hunt's sending off.
The resumption saw Singh equalise with a fierce left-foot shot which flew under Harper, before Bramble latched on to a pass from sub Jermaine Easter and rounded Harper to make it 3-2.
Northampton had chances to at least grab a share of the spoils, but Briggs secured victory for the home side with a sweet left-foot strike.