The Dutch return to the finals after missing out on the 2018 tournament in Russia but with little conviction after limping through their last two qualifiers.
They had been eight minutes from securing early qualification on Saturday before giving up two goals to draw in Montenegro, but Tuesday's victory in an empty stadium due to COVID-19 restrictions was enough to top the group.
Netherlands finished with 23 points from 10 games, two more than Turkey, who won 2-1 away to Montenegro to secure a playoff spot. Norway ended third in the group with 18 points.
Bergwijn hammered home a right-foot shot in the 84th minute that flew into the back of the net, to the relief of the home side. Depay made sure of the result as he finished off a breakaway after being set up by Bergwijn deep in stoppage time.
The Dutch dominated possession and had several half chances with Depay, their most enterprising player.
But they also played with an edgy approach, eager to avoid mistakes that would allow Norway to snatch the game away.
The Norwegians, without injured talisman Haaland, would have had to win to secure a top-two finish and show some enterprise in the last quarter, but they still failed to get any efforts on target. The absence of supporters did not help the home cause.
The game was played behind closed doors at De Kuip after the Dutch government over the weekend barred spectators from major sporting events for the next three weeks in an effort to contain a rapid surge in COVID-19 cases.
Netherlands coach Louis van Gaal was also missing from the dugout. He fractured his hip falling off his bicycle on Sunday and sat in a wheelchair in the VIP area, high in the stands, communicating via mobile with his assistants on the bench.