Sundowns moved the ball around strongly and comfortably and managed to find huge amount of spaces in defence.
Hlompho Kekana popped up inside the box and fired a shot away but missed the target. Andile Jali followed later on with a similar effort, but he too failed to find the target.
Brazilian-born Ricardo Nascimento pushed forward after finding a few yards of spaces and fired in a strong shot down the throat of Patrick Tignyemb.
The Cameroonian was working overtime and was called on to make a save at the near post with a glancing header from Toni Silva close to the midway mark.
At this stage, Celtic were making no threats but were looking dangerous on the break after managing to find openings on a few occasions.
The Free State side prevented the late pressure in the half to maintain a goalless score-line heading into the tunnel.
Sundowns missed a glorious chance from five metres with Themba Zwane shooting wide from a nice layoff by Silva.
Siwelele maintained their first half shape, continuing to frustrate the home side who were reduced to mostly shots from range.
On the hour mark, Downs pounced when Gaston Sirino ran onto a long ball to beat Tignyemb in a one-on-one scenario.
Steve Komphela’s charges woke up and changed their approach, piling more pressure in search of the leveller but dilly-dallying inside the box only worsened their situation.
In the final stages, Downs kept up the pressure and held on for the victory with little to no threat on the goal of Denis Onyango.
Phakamani Mahlambi failed to put the icing on the cake in added time with a wayward finish after being sent clear as the Tshwane giants book a spot in the next round.
Sundowns: Onyango, Ngcongca, Lebusa, Nascimento, Langerman, Jali, Kekana, Maboe, Zwane, Sirino, Silva.
Subs: Mweene, Mabunda, Madisha, Lakay, Mahlambi, Brockie, Meza.
Celtic: Tignyemb, Letlabika, Pfumbiza, Rikhotso, Mabena, Mashikinya, Masuku, Gordinho, Dlamini, Phalane, Letsoalo.
Subs: Dambe, Sam, Mabasa, Maema, Motshegwa, Shikweni, De Jongh.