Brayan Leon scored the only goal of the game with a powerful header from close range after 52 minutes.
Sundowns will now take a slender but important advantage into their second leg at home on Saturday, 18 April.
Esperance started brightly with Hamza Jelassi forcing Brazilians stopper Ronwen Williams into a point-blank save.
A long-range Brayan Leon effort then nearly caught out home keeper Bechir Ben Said but he composed himself to ultimately make an easy stop.
Kouceila Boualia made a storming run down the left flank for Esperance and put in a good cross that Williams claimed well.
Grant Kekana then needed all of his power to beat Florian Danho to the ball as the powerful French forward broke through on goal, before Khuluiso Mudau took a chance with a slide tackle in the box on Abdramane Konate that he timed to perfection.
Sundowns’ best chance arrived in the 28th minute when Arthur Sales was set up for a shot by Teboho Mokoena but his drive was deflected wide of the post with the keeper beaten.
The frame of the goal then came to the rescue of Sundowns as Danho raced through on goal from a quickly taken throw-in, and while he beat Williams, the ball slammed back into play off the crossbar.
Danho was in again after some excellent approach play down the right flank from Esperance but Kekana got back to make an excellent sliding tackle.
The Sundowns goal was under siege from the resultant corner and Williams was forced into a point-blank save off the head of Mohamed Tougai.
It had been a testing first half for the South African champions, but Downs were quickly out the blocks at the start of the second half and Leon so nearly found an opener with a diving header that Said saved well.
READ: Mamelodi Sundowns head coach Miguel Cardoso welcomes key duo back from injury
But there was no stopping Leon from heading home on the 50-minute mark to hand Sundowns the lead.
Jayden Adams swept the ball wide in the air to Thapelo Morena, who volleyed first time into the box where the Colombian marksman made no mistake from close range.
In response, Jack Diarra and Aboubacar Diakite were introduced just after the hour mark for Esperance as they looked for a way back into this game, with Yan Sasse and Boualia withdrawn,
Sundowns were then guilty of some erratic defending as they conceded their lead to a 66th minute free kick, before the decision was overturned a number of minutes later after a VAR review
Mohamed Ben Hamida had whipped a ball to the edge of the 18-yard-area and Williams came out of his goal to punch clear but was taken out by a combination of Sales and Jelassi, who was ultimately cautioned for an elbow into the face of the Sundowns stopper.
Play had initially been waved on, and Ibrahima Keita volleyed the loose ball to the far post where Konate stooped to head home.
Esperance continued to press forward in search of an equaliser but Keita couldn’t keep his header down, as Diarra fired in from the edge of the area, that Williams saved with ease.
There was no rest for Downs as the game wore on with the home side relentless in their search for an equaliser.
Sundowns were then dealt a significant blow with Kekana sent off for a boot to the face of his man, with five minutes of the match remaining.
Esperance were so nearly level from the resultant free kick but Ogbelu somehow headed the ball on the outside of the left upright.
Morena was withdrawn for Khulumani Ndamane and Leon came off for Zuko Mdunyelwa for the final three minutes of the match.
There was still time for the home team, with ten minutes of stoppage time added to the end of the match, but Sundowns were not to be beaten and now take a slender but vital advantage to their second leg at the Loftus Versfeld Stadium next week.