Kakko cut it to 2-1 at 12:17, batting in a floating puck from in front off a Filip Chytil pass attempt.
Kakko nearly tied it at 18:36 with a one-timer from the left circle but Levi got across and sprawled to make the save, prompting chants of "Le-vi! Le-vi!" from the crowd.
The chants continued, including early in the third period after Levi made a pad save on a Barclay Goodrow spin-shot from the slot and stretched the pad to deny Jimmy Vesey on the rebound.
"It was unbelievable," Levi said. "The fans were unreal. It's so nice having that support coming in as a new guy. It just makes it so much easier, to have a warm welcome from everyone. All the fans just like cheering your name, it's unbelievable. You know, I can't take credit for the win. The guys played a [heck] of a game also in front of me."
NOTES: Fox has nine points (two goals, seven assists) in his past 10 games. … Levi (21 years, 94 days) is the third-youngest goalie in Buffalo history to win his NHL debut after Tom Barrasso (18 years, 188 days) and Daren Puppa (20 years, 223 days). … Levi made 19 saves in the third period, tied for third-most in a period in Sabres history, behind Ukko-Pekka Luukkonen (21) and Ryan Miller (20). … Peterka scored for a second straight game, becoming the fourth Buffalo rookie in the past 10 years to score in consecutive games multiple times in a season. … Sabres defenseman Mattias Samuelsson played 22:13 and had three hits in his return after missing eight games with an upper-body injury.