But he pushed on, with his forces very much scattered, until Bragg's troops from Mississippi began to join him.

Then Bragg took the initiative.
Rosecrans had to fall back in turn, and was able to get his army together at Chickamauga, some miles south-east of Chattanooga, before the main battle was brought on.

The battle was fought on the 19th and 20th of September, and Rosecrans was badly defeated, with a heavy loss in artillery and some sixteen thousand men killed, wounded and captured.
The corps under Major-General George H.Thomas stood its ground, while Rosecrans, with Crittenden and McCook, returned to Chattanooga.

Thomas returned also, but later, and with his troops in good order.

Bragg followed and took possession of Missionary Ridge, overlooking Chattanooga.