While those baby mittens are awfully cute, and effective at protecting baby's cheeks, eventually you'll have to trim your baby's nails. The safest method of keeping nails short is a nail file (or emery board). Unlike baby nail clippers or scissors, these do not leave sharp edges that can still cut little faces.
If your baby is resistant to having their nails trimmed, you may need to wait until they fall asleep. Once they're in a deep sleep, it should be a bit easier to file the nails down. And, don't worry if your baby scratches their face. It happens to all of us.