Itumeleng Khune dropped to the substitutes' bench as Daniel Akpeyi returned in goal, while Ramahlwe Mphahlele replaced the suspended Philani Zulu.
Chiefs were off to a fast start when Anthony Akumu Agay spotted an opening in the centre of the area on nine minutes to head home Khama Billiat's shortly taken corner-kick.
Billiat pulled clear on the far side towards the midway stage and looked certain to double his side's lead after cutting into the inside left from out wide, but in the end curled over the top.
Siwelele Sa Masele regained control after the slow start and a few promising balls into the channels and six-yard area by Siphelele Luthuli and Menzi Masuku kept Chiefs on guard.
Chiefs should've doubled their lead in the last seven minutes of the half with a long-range strike by Njabulo Blom, but Jackson Mabokgwane went across the bottom-corner to parry.
Siwelele threw on four attacking changes at halftime in Motebang Sera, Tumelo Njoti, Ryan De Jongh and Tumelo Mangweni in pursuit of a comeback.
The Amakhosi still enjoyed the territorial advantage ten minutes from the restart and should've gone two goals up from a first-time strike by Leonardo Castro, but it rolled past the far corner.
The visitors levelled it up midway in the second half when Mangweni brought down a delightful pass from De Jongh to volley home shortly after Akpeyi smothered a header from Victor Letsoalo.
Chiefs struggled to regroup thereafter, allowing Celtic a lot of time and space on the ball with openings into promising area too.
In the end, it finished 1-1 with the Amakhosi's winless run of form in the league continuing down the bottom-half of the table.
Chiefs: Akpeyi, Ngezana, Cardoso, Mathoho, Mphahlele, Blom, Agay, Ngcobo, Manyama, Billiat, Castro.
Subs: Khune, Parker, Katsande, Moleko, Kambole, Ntshangase, Sasman, Ntiya-Ntiya. Zuma.
Celtic: Mabokgwane, Mahashe, Matjila, Baloyi, Lingwati, Phalane, Masuku, Luthuli, Letsoalo, Mabena, Potsane.
Subs: Chaine, Mashikinya, Chabalala, Ngobeni, De Jongh, Mangweni, Sera, Njoti, Ndwandwe.