There are actually two answers to this question. Ice Hockey is the fastest moving team game in terms of overall speed of play. Jai Alai is the fastest team game in terms of how fast the ball moves (it's been clocked at over 188 mph).
The fastest goal scored from the start of a period is 4 seconds, by Denis Savard in 1986. The fastest goal scored at the start of a game is 5 seconds, by Doug Smail in 1981, Bryan Trottier in 1984, and Alex Mogilny in 1991.