|Close, but not a debut.|
A select few actors and actresses have won Oscars for their debut performances. While twelve actresses and two (supporting) actors received Academy Awards for their first roles, no debut performance has won a Best Actor Oscar. The Academy gravitate towards people they know, so this makes perfect sense.
Besides, only one debut performance from the 2000s (Jennifer Hudson for Dreamgirls) and one from the 1990s (Anna Paquin for The Piano) have won. The other twelve wins occurred between the 1930s and the 1980s.
However, some men have come very close. Ben Kingsley (Gandhi) and Geoffrey Rush (Shine) scored wins for their first major performances, but not technically their debut ones. It doesn't seem like this trend will change, but maybe a new contender will emerge in the upcoming award season.