Remove any chunks of pizza with a paper towel without pushing down on the fabric, until there is only the sauce portion of the pizza left. If you are just discovering the pizza stain but it is old, use a butter knife to break up and scrape away the dried sauce to remove the solid portion.
Mix a solution of equal parts water and white vinegar. Dab the vinegar solution onto the fabric with a clean cloth to wet the fabric without saturating it to soaking wet.
Rub the fabric with the cloth using an even amount of pressure. Don't forcefully scrub the fabric so hard that you distort the fabric and actually push the stain further into the fibers.
Dab another small amount vinegar solution on the fabric and allow the chair to sit undisturbed for 5 to 10 minutes to really strip the stain from deep in the fabric.
Use a second clean cloth and press it into the fabric to soak up the vinegar solution until the previously stained area only feels damp. Check the stain to make sure it's gone, and repeat Steps 2 through 5 until the stain is stripped from the upholstery.
Dab plain water onto the chair with a sponge to dilute and bring up the remaining vinegar mixture. Press with a dry cloth until the fabric feels only damp. Allow the chair to dry thoroughly on its own, or place the chair in a sunny spot to dry a little faster.