Witnesses said the clashes erupted after police pulled down the tents of about 100 protesters who had camped in the square overnight in an attempt to restart a long-term sit-in after a mass demonstration Friday against the country's military leadership.
Crowds swarmed an armored police vehicle, setting it on fire and throwing stones at police.
Protests also took place in other cities, including the northern port city of Alexandria, where at least one other person died, and in the eastern city of Suez on the Red Sea.
The military-controlled government called on protesters in Tahrir Square to clear the area. Egyptian state television reported that police arrested at least 18 people, describing them as rioters.
The unrest was reminiscent of the anti-government protests that forced an end to former President Hosni Mubarak's rule in February.
The violence raised fears of new unrest before Egypt's first parliamentary elections since Mr. Mubarak's fall. The vote is scheduled for later this month.
Tens of thousands of protesters jammed the square on Friday in reaction to a government proposal that would exempt the military from civilian oversight in the next constitution. Critics called the plan a military attempt to prolong its "temporary" powers.
The rally was called by the Muslim Brotherhood, the country's most organized political group.
Egyptians begin going to the polls on November 28 for parliamentary elections that will stretch into March.
The new legislature will draft a constitution ahead of presidential elections.