You can place the Cruet in the dishwasher.
However, the dishwasher jets may not wash the inside of the two vessels very well. In that case, take a small piece of cloth, add dishwasher soap to it, and put it inside one side of the Cruet at a time. Add a small amount of warm water. Place your finger over the spout and shake well. The wet cloth will shake around to all sides of the glass, cleaning it.
Rinse and repeat if necessary. You can remove the wet cloth with a toothpick or tweezers to get it back out of the spout.
After thorough clean water rinsing, you can dry the Cruets in the oven at 350 degrees. All condensation will evaporate in just a few minutes. Let cool before touching. The Cruet is made from Pyrex glass, a laboratory technical grade glass that will not break from heat of 350 degrees.