Olofsson made it seven straight on a one-timer off Rasmus Dahlin's pass from the right circle to open the scoring 6:27 into the second period against Dallas on Monday.
Nicknamed "Goal-ofsson" by Sabres fans, the 24-year-old Swedish player has scored five goals in Buffalo's first six games. His streak dates to the end of last season, when he scored twice in six games.
The Sabres selected Olofsson in the seventh round of the 2014 draft. He spent five years playing in the Swedish Elite League before signing with the Sabres in April 2018.
Three players shared the previous record for scoring their first six career goals on the power play: Winnipeg's Craig Norwich in 1979-80, Hartford's Sylvain Turgeon in 1983-84 and Islanders' Jeff Norton in 1987-88.