Turn off the water to the irrigation system.
Dig a hole straight down into the ground at the wet spot, using a shovel. Or if it's beside a riser or pipe with a sprinkler coming out of the ground, dig around it. Make the hole at least 24 inches wide. When you get down to about 18 inches deep, slow down and dig carefully until you make contact with the horizontal irrigation pipe. Use your hands to clear away dirt from the pipe until you can see it clearly, and use a rag to wipe away dirt from the pipe's surface.
Turn on the water. Watch and listen carefully for signs of leaks along the exposed section of pipe. It should create a small shower, trickle or hissing sound where the water is escaping under force. When you locate the leak, turn off the water and let it drain from the hole.
Cut off the pipe on both sides of the leak, with a hacksaw. Or if it's leaking from a fitting, cut off the fitting, take it to a home improvement store, and buy a new one that's the same size and type. If you've cut off only a straight section of pipe, buy a replacement piece and two couplers that are the right size to fit your pipe. Typically, irrigation pipe has a 1 1/4 inch diameter.
Replace the section that you cut out, using PVC glue. If it's just a straight section of pipe, glue the couplers to both ends.
Clean the pipes that are in the hole, where you cut away the pipe. Use a wet cloth to get all the dirt off, and let them dry thoroughly. Spread glue on both ends that you cut off. Lift the pipe up gently, as high as you can without breaking it. Insert the new section or fittings onto both ends of the pipe, and drop it down into the hole. As the pipe lowers, it will tighten the couplers. Wait at least 1 hour for the glue to dry.