Shalulile became the leading scorer in all domestic competitions this season with 15 goals after scoring the winner on 17 minutes in eastern city Mbombela.
An attempted headed clearance by Galaxy off a corner came into the path of the Namibian, who nodded past goalkeeper Wensten van der Linde for his 11th Premiership goal. Shalulile could have had a hat-trick by half-time and was a constant threat to mid-table Galaxy, a second division club last season who bought the franchise of disbanded Highlands Park.
Mweene came to the rescue of Sundowns 15 minutes later by diving to his right and saving a weakly struck penalty from Lindokhule Mbatha. The Zambian was deputising for first choice Dennis Onyango, who travelled to Uganda for a ceremony in his honour after recently retiring from international football.
Sundowns need two victories from their remaining four fixtures -- two at home and two away -- to clinch a record fourth straight Premiership title. They have 57 points after 26 rounds, seven more than AmaZulu, the Durban club transformed from perennial strugglers into title contenders by coach and former South Africa star Benni McCarthy.
Sundowns now adjust their sights to the CAF Champions League as they face title-holders Al Ahly of Egypt on Saturday in a quarter-final first leg in Cairo.
Standout matches today