1. Fill the 5-gallon jug and use it to fill the 3-gallon jug. That leaves 2 gallons in the 5-gallon jug.
2. Empty the 3-gallon jug and transfer the 2 gallons from the 5-gallon jug to the 3-gallon jug.
3. Fill the 5-gallon jug and use it to top off the 3-gallon jug. That requires exactly 1 gallon, leaving 4 gallons in the 5-gallon jug.

You can also solve this geometrically, or you can let John McClane solve it for you:

04/23/2017 UPDATE: Two readers give a more efficient solution:

1. Fill the 3-gallon jug and empty it into the 5-gallon jug.
2. Fill the 3-gallon jug again and use it to top off the 5-gallon jug. This leaves 1 gallon in the 3-gallon jug.
3. Empty the 5-gallon jug. Transfer the remaining gallon from the 3-gallon jug to the 5-gallon jug.
4. Fill the 3-gallon jug again and pour it into the 5-gallon jug.

This solution requires discarding 5 gallons altogether, where mine required discarding 6. (I think this is probably the alternate solution that Grossman’s method will obtain, but I haven’t had time to check.)

(Thanks, Chris and Marty.)