The only goal came inside three minutes in northeastern city Mbombela when a looping header from Masilake Phohlongo beat goalkeeper Siyabonga Mpontshane.
It was the third league goal for the midfielder this season, and one of the other two also came against Pirates in a draw in Soweto last January.
A bad-tempered first half ended with both teams reduced to 10 men as Abel Mabaso of Pirates and Galaxy captain Given Msimango were sent off after being yellow-carded twice.
Losing to mid-table Galaxy left Pirates four points adrift of second-place AmaZulu with only two rounds remaining in the richest African league with a 15 million rand ($1.1 mn/895,000 euro) first prize.
Champions Mamelodi Sundowns, who sealed a fourth straight title last Wednesday, and the runners-up qualify for the Champions League, and the side finishing third goes into the second-tier CAF Confederation Cup.
Pirates lie fourth, beneath Golden Arrows on goal difference, and the two clubs are set to fight for the Confederation Cup spot as one victory will ensure AmaZulu go into the marquee African club competition.
The Buccaneers had beaten AmaZulu three days ago thanks to a brilliantly converted free-kick from Malawian Frank ‘Gabadinho’ Mhango with 14 minutes left in Soweto.
“We have to win our remaining two matches to have a realistic chance of a top-three finish and qualification for Africa,” said Pirates’ German coach Josef Zinnbauer after the defeat by Galaxy.
“Another very difficult away match awaits us on Wednesday against Baroka and we finish our campaign at home to TTM, who could be fighting to preserve their Premiership status.”
Many Pirates supporters took to social media this week calling for Zinnbauer to be sacked after a 5-1 aggregate loss to Raja Casablanca of Morocco in a Confederation Cup quarter-final last weekend.
On Saturday, 2016 African champions Sundowns drew 0-0 away to fifth-place Swallows in Soweto to move 11 points clear of AmaZulu.
It was a predictable outcome as promoted Swallows have drawn a record 19 matches this term — an unwanted Premiership landmark previously held by Free State Stars, Santos and SuperSport United with 17 each.
Drawing kept the 14-match unbeaten home record of Swallows and the 14-match unbeaten away record of Sundowns intact.