The Brit appeared to get a good start, fending off team-mate Oscar Piastri and George Russell into Turn 1, before Piastri got a better run heading into Turn 2.
Piastri overtook his McLaren team-mate for the lead, with Norris taken by surprise and also losing out to Ferrari driver Charles Leclerc, who moved up to second.
Meanwhile, world champion Verstappen called for a penalty for his fierce rival Hamilton, after accusing the Brit of not leaving enough room when attempting a pass.
Verstappen begins Italian GP charge
Following a poor qualifying where he could only put his RB20 in seventh, Verstappen started Sunday's race desperate to limit damage to Norris in the drivers' championship.
Norris started the race 70 points behind Verstappen in the championship, but with six places between himself and the Dutchman on the track.
Verstappen made up a position on the first lap, overtaking one Mercedes in Russell as the Brit appeared to be forced off the track heading into Turn 1 and taking to the escape road, before the Dutchman attempted to do the same to old rival Hamilton.
However, the four-time world champion was left frustrated by his inability to pass the 39-year-old, taking to team radio to call for a penalty for Hamilton.
"Lewis didn't leave a car's width," Verstappen suggested. "I mean, I got a penalty for that.. so..."
Norris and Verstappen's contrasting starts will give Red Bull hope that they can extend their advantage in the constructors' championship standings.