Turkey's 2-1 win against Montenegro kept the pressure on the Netherlands, but they still needed only a point to secure their qualification and got more than that in the closing stages at De Kuip.
Bergwijn netted with six minutes left when he fired high into the net before Depay put the result beyond doubt with a goal on the break in injury time.
While the Oranje finished as Group G winners, Norway slipped to third and missed out on a place in the play-offs having failed to register a shot on target across the entire contest.
The Netherlands wrapped up top spot in Group G on Tuesday after a cagey 2-0 victory over Norway thanks to late goals from Steven Bergwijn and Memphis Depay.
Netherlands head coach Van Gaal had suffered a fall from a bike and injured his hip ahead of the game so was forced to watch the match from the stands in a wheelchair.
Van Gaal cut a nervous figure as Danny Blind relayed his instructions via telephone on the touchline in a fixture played without any fans at De Kuip due to growing coronavirus cases.
Despite a tense game, the Netherlands got the job done as their stubborn defence – which has conceded just eight goals in qualifying – stood firm.
The Netherlands are now unbeaten in seven matches since Euro 2020, winning five times.
"The mood was a bit down the day after [the draw with] Montenegro, then I was afraid that it might go completely wrong," Van Gaal told NOS Journaal.
"But all in all it was a fantastic week despite everything. We got to know each other through lows and highs.
"We scored two goals today, and Jasper Cillessen kept a clean sheet. We dominated and controlled the game for 90 minutes. Norway may have had half a chance.
“I complimented the team at half-time, but also told them to play a little more forward.
"We played provocative pressing in the second half, which gave us a little more space. And of course we have the attackers for that."
Netherlands captain Virgil van Dijk made a team-high four clearances, two tackles and won 88.9 per cent of his duels as Norway failed to register a single shot on target.
The Liverpool defender spoke of his excitement at potentially playing at his first major international tournament, having missed out on the rescheduled Euro 2020 earlier this year through injury.
"We are going to Qatar and it's well deserved, we almost didn't give anything away. In my opinion, nothing at all," said Van Dijk.
"We had to show a reaction after last Saturday and we did. Of course there are many things that could be improved, but the result was the most important.
“I am extremely proud of this team. It can be tricky to play around with the idea that any mistake can be fatal, but we've embraced it.
"We showed real character, it was not easy to switch quickly after the 2-2 in Montenegro, but we did it.
"I was very sorry that the national coach was there from a distance and the fans were not there at all. We're going to celebrate well inside. I can't wait for the World Cup."
The Oranje wrapped up top spot in Group G on Tuesday after a cagey 2-0 victory over Norway thanks to late goals from Steven Bergwijn and Memphis Depay.
Head coach Van Gaal had suffered a fall from a bike and injured his hip ahead of the game so was forced to watch the match from the stands in a wheelchair.
He cut a nervous figure as Danny Blind relayed his instructions via telephone on the touchline in a fixture played without any fans at De Kuip.
Despite a tense game the Netherlands got the job done as their stubborn defence – which has conceded just eight goals in qualifying – stood firm.
The Netherlands are now unbeaten in seven matches since the European Championships, winning five times.
"The mood was a bit down the day after [the draw with] Montenegro, then I was afraid that it might go completely wrong," Van Gaal told NOS Journaal.
"But all in all it was a fantastic week despite everything. We got to know each other through lows and highs.
"We scored two goals today, and Jasper Cillessen kept a clean sheet. We dominated and controlled the game for 90 minutes. Norway may have had half a chance.
“I complimented the team at half-time, but also told them to play a little more forward.
"We played provocative pressing in the second half, which gave us a little more space. And of course we have the attackers for that."
Netherlands skipper Virgil van Dijk made a team high four clearances, two tackles and won 88.9 per cent of his duels as Norway failed to register a single shot on target.
The Liverpool defender spoke of his excitement at potentially playing at his first major international tournament, having missed out on the rescheduled. Euro 2020 earlier this year through injury.
"We are going to Qatar and it's well deserved, we almost didn't give anything away. In my opinion, nothing at all," said the Liverpool defender.
"We had to show a reaction after last Saturday and we did. Of course there are many things that could be improved, but the result was the most important.
“I am extremely proud of this team. It can be tricky to play around with the idea that any mistake can be fatal, but we've embraced it.
"We showed real character, it was not easy to switch quickly after the 2-2 in Montenegro, but we did it.
"I was very sorry that the national coach was there from a distance and the fans were not there at all. We're going to celebrate well inside. I can't wait for the World Cup."
The 28-year-old revealed his regret at leaving Inter for Chelsea and questioned boss Thomas Tuchel's tactics, leading to him being dropped for the 2-2 draw with Liverpool.
Lukaku has returned for the Blues' two matches since, but doubts remain over whether he will still be a Chelsea player come the start of next season.
TOP STORY – LUKAKU LINED UP AS MBAPPE REPLACEMENT?
PSG look certain to lose Kylian Mbappe at the end of the campaign as the Real Madrid target fast approaches the end of his contract at the Parc des Princes.
The Ligue 1 giants are on the lookout for possible replacements and, according to El Nacional, Lukaku is one of those being considered by the club's wealthy backers.
The report suggests PSG are willing to pay €90million (£75.1m) to persuade Chelsea owner Roman Abramovich to part ways with the club's all-time record signing.
Lukaku, who apologised for giving the explosive interview in December, has scored eight goals in 20 appearances since returning to west London five months ago.
ROUND-UP
– Tottenham have decided to sell Steven Bergwijn this month amid interest from several clubs, according to Fabrizio Romano. Ajax are said to be among the sides to have already made an offer for the Netherlands international.
– Anthony Martial wants to leave Manchester United and has been touted as a possible target for Barcelona. But Spanish outlet Sport claims that, while Barca are eager to strengthen in the final third, Martial is not on their radar in this window.
– Chelsea defender Antonio Rudiger is wanted by a number of Europe's heavyweight clubs. According to Foot Mercato, PSG are prepared to pay more than the likes of Bayern Munich and Real Madrid, but the latter remains the Germany international's preferred choice.
– A rumoured target for Chelsea and Bayern, Sergino Dest's Barcelona future may become clearer in the coming days. El Nacional claims the defender's representatives are set to hold talks with the Catalan giants.
– After being jeered off the pitch by his own supporters in Tottenham's 3-1 FA Cup win over Morecambe on Sunday, The Athletic reports that club-record signing Tanguy Ndombele is hopeful of ending his tumultuous spell in north London this month.
– The Athletic also reports Aston Villa are now the favourites to sign Everton full-back Lucas Digne, despite previous interest from Chelsea. Newcastle United have also been linked, but Digne is apparently not considering a move to the relegation-threatened side.
