Midfielder Mount opened the scoring on eight minutes with a drilled shot from the edge of the 18-yard box into the bottom left-hand corner as the visitors struggled to keep pace with the league leaders.
Chelsea's advantage doubled minutes later as winger Callum Hudson-Odoi, who was making his third league start of the season, slotted past Norwich goalkeeper Tim Krul. Defender Reece James added a third just before the break, as he careered down the right-hand side before chipping home from 6 yards out.
Thomas Tuchel's side continued to dominate in the second half, and defender Ben Chilwell took advantage, scoring for a third consecutive Premier League game with a left-footed strike from the edge of the 18-yard box.
Chelsea were awarded a penalty on 82 minutes for a handball that Mount took but was saved by Krul. However, the penalty was retaken after it was shown that the Dutch goalkeeper had encroached on the line.
Mount made no mistake with his second attempt and powered it past Krul. He added his third deep into added time with a shot straight down the middle of Norwich's goal.
"Before the game, I put pressure on myself to get on the scoresheet, then three came around in the same game. It's obviously a massive moment for me. I'll remember this one for a long time," Mount said after the game.
"I was always getting in positions to score, but I just wasn't putting it in the back of the net. I thought the last one might be offside. It was my luck today, but it's also about the team. We spoke before about being at it from the beginning -- we did that today.
"It's a big moment for me to score three at home at Stamford Bridge."