Substitute Lehlohonolo Majoro blasted home a rasping left footer for the third goal for the Buccaneers.
All goals came from classic counter-attacks as Pirates caught their hosts with swift balls from deep and gave them a lesson in finishing.
It was Bucs' first league win and left the Brazilians still awaiting a home success this season.
The game ended on a sour note for Pirates as Thabo Matlaba was sent off for a stamping incident, but it detracted little from the way they engineered a massively morale-boosting win.
The first goal was a classic counter as Pirates won possession from a Sundowns attack, and two passes from Thandani Ntshumayelo and Happy Jele saw Erasmus bear down on goal and then scoop the ball with unerring accuracy over the head of disbelieving goalkeeper Kennedy Mweene.
Erasmus' brilliant 14th- minute goal was followed by a flurry of chances at either end as the opener enlivened the tempo of the game to the delight of the 20 000-plus crowd, who had delayed the kick off by 15 minutes with their late arrival at the venue.
In the space of three minutes, the ball struck the woodwork three times.
Hlompho Kekana blasted a long-range effort against the Pirates crossbar in the 27th minute.
But immediately the visitors counter-attacked, with Oupa Manyisa ghosting into the penalty area with impeccable timing, and took advantage of Alje Schut's lapse of concentration to toe-poke the ball against the left upright of Sundowns' goal.
A minute later, Surprise Moriri plucked a ball out the sky onto his chest and then hit a thumping full volley in almost the same place Kekana had struck just seconds earlier.
Again to the dismay of the Sundowns' fans, the shot cannoned back into play off the crossbar.
Pirates' short-stopper Senzo Meyiwa made a fine stop to deny Teko Modise after a clever one-two with Bongani Zungu.
Pirates could have been 2-0 up when Erasmus beat the two Sundowns' central defenders and set himself up from close range, only to hit agonisingly wide.
Meyiwa then made two potentially calamitous errors but recovered on both occasions. The first came as Mame Niang almost catch him in possession, while the second saw him unsure of how to handle a dipping free-kick, which he eventually scrambled away with his feet.
After the break, Downs took the game to Pirates, who countered again in the 63rd minute as Daine Klate set Erasmus away, and again he lofted the ball over a bemused-looking Mweene.
Erasmus was then replaced by Majoro, who hammered home a left-footer for his third goal in two games after another counter attack that Thabo Nthethe did not clear properly.
The Buccaneers protested before the game over an outdated registration card for Schut, but after winning they might drop the action.
l Free State Stars pipped University of Pretoria 1-0 at Tuks Stadium in Tshwane in another Premiership match last night.