Bergwijn, the standout player of the evening on the right wing, drilled the ball into the roof of the net following an assist by Arnaut Danjuma after a lacklustre 83 minutes that saw both sides struggle to create chances.
Bergwijn again played a key role in the Dutch's second goal a few minutes later, after a counter-attack saw him run almost the length of the field before passing the ball to Depay, who slotted it past a diving Orjan Nyland.
Depay's 12th goal of the qualifying campaign rounded off the evening for the Netherlands, who only needed a draw against Norway to finish top of Group G ahead of Turkey and the Norwegians.
"They stuck to the game plan, they were a lot more compact," Dutch coach Louis van Gaal told the NOS public broadcaster.
Van Gaal admitted he was "worried that things could go wrong", especially after the his side's shocking 2-2 draw against Montenegro over the weekend when they let a two-goal lead slip in the final 10 minutes.
"But this evening we kept the match under control," he added.
Both teams seemed indecisive at times in cold and rainy conditions at De Kuip and the match seemed set for the defences to come out on top.
Throughout the game the Oranje created more chances and Norway never really threatened the Dutch goal, manned by veteran 'keeper Jasper Cillessen.
The hosts played with an added disadvantage after Van Gaal injured his hip after a fall on Sunday and was confined to a wheelchair up in the stands.
This saw Van Gaal constantly relaying instructions to his technical team next to the pitch.
The first real opportunity for the Netherlands came when the hard-working Danjuma saw a strike slide wide in the 26th minute.
This was followed by several shots and headers at goal by Depay, but all of whom went into the hands of Norwegian goalkeeper Nyland.
Stale Solbakken's men clearly missed the presence of star striker Erling Haaland, sidelined because of injury.
Norway never troubled Cillessen and the closest they came was a desperate shot into the stands in the 88th minute.
Dutch captain Virgil van Dijk praised his team, saying: "It was a solid win, we almost gave nothing away."
Barcelona forward Depay said it was "an emotional game".
"We played with everything we had today and we are going to Qatar," he said.
Standout matches today