On the 2001 4Runner the DRL relay is inside the vehicle, near the driver's right leg, just left of the center console. It is labeled "Running Lights." You can reach it without taking apart the dash at all; just reach behind it and unplug the connector, remove pin 2, tape it securely out of the way, and plug the connector back in.
I have a 2000 4Runner which has a grey box that is unlabled. I followed the instructions above and it worked. To assist any who attempt to remove PIN2 please note you will want to find a very small flathead screwdriver. I actually grinded a small screwdriver down very flat so I could push it into the backside of the PIN2 hole which relieved the catch inside the connector. Once I had the small screwdriver inserted into the backside of the PIN2 hole I pushed backwards from the front and PIN2 slipped out. This way if you decide to replace PIN2 all you have to do is snap it back in.