I'm defining the first round as the first round played in a given tournament, excluding Opening Round/First Four games. As far as I can tell, the only other team that is undefeated in their first NCAA Tournament games is Tulane, who won their first game in 1992, 1993, and 1995, and lost their second game all three years and haven't made the tournament since.
As for teams that won their first three or more first round games, this is what I found:
First 12: Maryland (they are 27-3 in their first tournament games!)
First 8: Oklahoma State
First 7: California
First 5: Seton Hall
First 4: Lamar, Long Beach State, Michigan State, Notre Dame, Ohio State
First 3: Bradley, Canisius, Davidson, Drake, Duquesne, Georgia Tech, James Madison, Northeastern, Northwestern, San Francisco, Santa Clara, St. Bonaventure, Syracuse, Tulane, UNLV
(I'm having a wild Saturday night if you couldn't tell)