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…
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…