They briefly occupied one bridge and then marched to parliament after a meeting, carrying torches and shouting "The University is Ours" and "Free University".
They demanded the resignation of education secretary Rozsa Hoffmann and, escorted by police, later blocked another central bridge for hours. The march remained peaceful.
"The government has announced out of the blue that they will drastically cut the number of state-financed places in higher education, to 10,000 from next year," said Laszlo Bernath, 26, a history student.
"So Hungarian higher education will practically be transformed into a place churning out rich idiots."
The government of Prime Minister Viktor Orban, struggling to keep the budget deficit below the EU's ceiling of 3 percent of output, announced last week that from next year it would fully finance only 10,480 students, two-thirds fewer than this year.
The government will partly finance the tuition of 46,000 more and offer subsidized preferential-rate student loans.
Students in Budapest are planning to hold another protest on Wednesday, according to their Facebook page. Earlier on Monday students in the city of Szeged in the southeast of Hungary staged a small-scale sit-in.
"The reform of higher education is aimed at creating a high-quality, sustainable system for the long term," said a statement on the government's official website.
"We understand if students feel that they would like to express somehow that they are afraid of the changes, but we trust that this protest will turn into understanding, because changes are for them and not against them."
(Reporting by Krisztina Than; editing by Andrew Roche)