Goals from Nyasha Mushekwi and Elias Pelembe secured the victory for the Brazilians, taking them five points adrift of leaders Ajax Cape Town with a game in hand.
Amakhosi are six points back from the top with just two rounds of the season remaining.
The match started at a furious pace, and Chiefs created better chances in the first half; three times they broke the Sundowns offside trap but they were unable to capitalise on the opportunities.
Mandla Masango had two of the chances, first being brilliantly challenged by makeshift right-back Siyabonga Ngubane, the second time curling his shot wide of the far post.
Josta Dladla also went one-on-one with Waynde Sandilands after a great pass from Mthokozisi Yende but was denied by the goalkeeper.
Just past the half-hour mark Sundowns broke the deadlock.
Chiefs were unable to clear their lines after repeated attempts on goal from Teko Modise and Katlego Mphela.
Eventually Mphela guided the ball away from traffic in the middle of the box towards Nyasha Mushekwi, who took advantage of poor positioning from Itumeleng Khune to score with a left-footed shot.
And just seven minutes into the second stanza, the Brazilians doubled their lead.
Mphela was again the architect, providing a looping cross from the right of the penalty area, and Pelembe crashed home an unstoppable volley.
Chiefs reacted by using all three substitutions within 11 minutes of going 2-0 down and they proceeded to dominate the remainder of the game.
However, they were unable to find a way past Man of the Match Sandilands, who made several excellent saves.
Sundowns could have even added to the scoreline through Mphela, who found himself one-on-one with Khune, but his chipped finish drifted just wide of the post.