Vilardi extended the lead to 3-1 at 14:40, chipping in a one-timer from the left circle.
Pierre Engvall cut it to 3-2 at 16:58, but Fiala put the Kings back in front 4-2 at 2:11 of the third period, taking the puck from Rasmus Sandin in the neutral zone and scoring on a breakaway.
"We didn't get off to a great start last year," said Tavares, who has five points (three goals, two assists) during his streak. "It's not what we were looking for this year. You obviously want to get off to a good start, and you play to win every game. Every game we've had a chance to win. Have we played to our capabilities yet? I don't believe so, so we're going to stay with it, keep working."
NOTES: Kings forward Rasmus Kupari made his season debut after being recalled from Ontario of the American Hockey League on Saturday. He was plus-1 and won 3-of-9 face-offs in 8:22. … Maple Leafs defenseman Filip Kral had one shot and was plus-1 in 11:48 in his NHL debut. He is the first player in Toronto history to wear No. 82.