Seventh seed Ruud won the first set against world number 41 Borges but appeared to be hampered by a leg injury as he slumped to a 2-6, 6-4, 6-1, 6-0 defeat.
Borges is through to the third round at Roland Garros for the first time and will play Australian 25th seed Alexei Popyrin for a place in the last 16.