AmaTuks had taken a 2-0 first half lead through Denver Mukamba and Geofrey Massa, but second half strikes from Bevan Fransman and Lerato Lamola ensured a share of the spoils.
Gregory Maasdorp had the first shot at goal, but blazed his seventh-minute shot well over the crossbar.
Mpho Matsi thought he had scored shortly afterwards, only for his header to be cleared off the line.
But Tuks did get their goal on 17 minutes when Man of the Match Mukamba netted from a corner kick.
Massa’s free-kick then shaved the crossbar, but the Ugandan did get his goal on 27 minutes, sending a thunderbolt past Patrick Tignyemb in the Celtic goal.
Matsi was stretchered off with a head injury, and the time added on almost resulted in a third for the visitors, only for Massa’s follow-up from Mukamba’s saved shot hitting the crossbar.
Celtic were a different team after the break and pulled one back on 51 minutes when Bevan Fransman buried Lyle Lakay’s free-kick.
Seven minutes later and it was all square – Lamola getting his tenth of the season from Hellings Mhango’s pass.
Mhango and Musa Nyatama came close for the revitalized Celtic, who also saw Lamola’s late shot from close range saved by Washing Arubi.
The draw moves Celtic up to fifth place, while Tuks remain in 10th place.