Yes, diamonds can cut glass.
Traditionally, in the lack of right tech-advanced tools, that's how craftsmen would cut glasses to make showpieces and other items.
There would be a pen with a sharp diamond at the tip.
(Courtesy: DH resource)
Such pens are still used but very rarely. Now, we've got the cutters and many other tools…
(Courtesy: Makita)
Anything that's harder (and more durable) than the glass can cut it.
On the Mohs scale, in terms of hardness, glass is rated between 6-7. In comparison, diamond is much harder, scoring a 10.
So, yes, you can use a diamond to cut glass.
It is, in fact, a popular way to identify the
authenticity of diamonds. If they can cut the glass, they are real. If not, they are fake.
However, this isn't a very reliable technique to find whether a diamond is real or fake. A lot of synthetic gems today, too, can cut glasses.
All said, here, check out this video of diamond cutting glass…