Dig up the area around the leaking T-fitting to expose the pipes. You can locate a broken fitting by running your sprinklers for 10 minutes and searching for sprinkler heads with water pooled at the base. Remove enough dirt so that your hand can work freely around the entire fitting.
Saw off both ends of the supply pipe at the end of the existing T-fitting. Use a hacksaw for best results, and cut the pipe as close to the fitting as possible.
Apply a thin layer of PVC cement to the inside of both ends of an extended-size T-fitting. The extra room on an extended adapter will allow you to connect to the existing pipes.
Press one end of the new T-fitting onto one end of the supply pipe. Gently lift up the fitting and the other end of the supply pipe until the second half of the supply pipe slides into the open end of the fitting. Release both the adapter and the pipe so they sit straight underground with the riser hole facing straight up.
Insert a sprinkler riser into the hole on the top of the T-fitting, turning it clockwise until tight.
Insert a sprinkler head onto the other end of the riser, and turn it clockwise until tight.
Replace the dirt around the pipe and sprinkler head until the surface of the dirt is flush with the surrounding ground.