The Senate’s party-line 21-7 vote Monday sends the bill to the House.
The bill originally would have allowed people with permits for concealed weapons to carry the weapons anywhere on campuses, including inside buildings.
But it was watered down last week.
Supporters of the original proposal say students and faculty members with weapons could defend themselves from attack while waiting for police to respond.
Opponents say allowing guns on campus would create dangerous situations in which officers responding to a shooting might be confused about which armed person was the wrongdoer.