The series leader and three-time world champion delivered a home win at the Red Bull Ring with a fighting response after being passed by McLaren's Norris in the opening laps.
His win was his third sprint triumph this year, his fifth in succession and his 10th in 15 contests as he came home 4.616 seconds ahead of McLaren's Oscar Piastri and 5.348 clear of Norris, who was unable to maintain his early speed.
"I had a good first lap," said Verstappen. "But once the DRS opened it took a few laps for me to get away and then I drove my own race. I had to work for it in that race and we have a few things to think about for tomorrow."
George Russell finished fourth for Mercedes ahead of Ferrari's Carlos Sainz and seven-time champion Lewis Hamilton in the second Mercedes. Charles Leclerc in the second Ferrari was seventh.
Sergio Perez came home eighth in the second Red Bull ahead of Haas's Kevin Magnussen and Lance Stroll of Aston Martin.
Verstappen was given a rousing reception by the massed ranks of his 'orange army' in the grandstands, notably roaring with delight when he responded to snatch back his lead from Norris.
On a hot day in the Styrian Alps, Verstappen made a strong start from pole, after two formation laps, to lead through Turn One ahead of Norris with Piastri hustling his team-mate – and Sainz passing Russell for fourth.
The whole grid started on medium tyres, hoping to survive the rising temperatures and complete the distance of 24 laps, reduced to 23, without alarm.
The McLarens stayed in close proximity and threatened not only each other but the leader as they searched for space to pass, all three cars separated by less than six-tenths of a second.
Having made several moves, Norris finally found a way past the Dutchman at the top of the hill on lap five, but Verstappen responded immediately and regained the lead at the next corner allowing Piastri to use his momentum to take second.
Norris, who was feeling unwell before the race, was left struggling to resist the charging Sainz while Piastri set off in pursuit of Verstappen.
By lap eight, the Mercedes were producing a measured helping of pace and this allowed Russell to pass Sainz on the outside of Turn Four for fourth with the help of Drag Reduction System (DRS).
Hamilton, in sixth, closed steadily on Sainz, who clung on while his Ferrari team-mate Leclerc rose from 10th to seventh, but was unable to make any further impression.
At the front, Verstappen relaxed after regaining control and moved out of DRS reach for Piastri who, by lap 15, was looking in his mirrors to see team-mate Norris closing in again. Verstappen led by two seconds.
As it unfolded, it was Verstappen controlling affairs and opening up a gap of more than four seconds as he cruised home before turning his attention to preparations for Grand Prix qualifying taking place later Saturday.
