Craig Bellamy's side completed an unbeaten six-game campaign and overhauled Turkey - 3-1 losers in Montenegro - at the top of Group 4 to secure elevation to League A.
That almost certainly handed Wales a World Cup play-off place to fall back on should they not automatically qualify for the 2026 tournament.
"We want to go to a World Cup. We've got to get that ticked off. If we get there, we want to compete," Bellamy said.
"What I liked was when we went 1-0 down we were able to play and we moved the ball."
Promotion was achieved the hard way following Andri Gudjohnsen's early strike - the first time Wales had fallen behind under Bellamy - and for 25 minutes the Welsh were staring at a relegation play-off against a League C runner-up.
Swansea City striker Cullen, winning his seventh cap, turned the tide with a first-half double before Brennan Johnson and Harry Wilson crowned victory with wonderful strikes.
Brennan Johnson scores his ğ�™‰ğ�™„ğ�™‰ğ�™�ğ�™ƒ goal of the season and it puts Wales in the box seat for promotion ğŸ�´ó �§ó �¢ó �·ó �¬ó �³ó �¿â€” Optus Sport (@OptusSport) Watch 📲 https://t.co/BqAA7I0EmF#NationsLeague #UNL pic.twitter.com/pI7NGW0mndNovember 19, 2024
"I probably feel it was important to show people I am not a lunatic, I'm quite sane," added Bellamy, whose temperament during a colourful playing career at the top level often got him into trouble.
"People thought I was going to be running on the pitch and pushing the ref and stuff, getting sent off.
"Trust me, you get nowhere being that way. I'm not naive enough to know those moments aren't coming."
In Budapest, Dominik Szoboszlai scored a late penalty to deny Germany a victorious end to Group A3.
Germany, already assured of a quarter-final spot, were on course to win thanks to Felix Nmecha's goal but Mihaly Kata's shot ricocheted off Robin Koch's arm as the German defender tried to turn away.
After a video review, Liverpool midfielder Szoboszlai stepped up to take the penalty and chipped it over goalkeeper Alexander Nuebel to salvage a 1-1 draw for Hungary, who finished third in the group.
The Netherlands finished second after a 1-1 draw at Bosnia-Herzegovina.
The last-eight line-up was already decided before Tuesday's games kicked off, with Portugal, France, Germany and Spain as group winners and Croatia, Italy, the Dutch and Denmark qualifying as runners-up. The draw is Friday at UEFA headquarters in Switzerland and the quarters will be played in March.
Elsewhere, the Czech Republic earned promotion from Group B1 after they beat Georgia 2-1 and second-placed Ukraine saw off Albania by the same scoreline.
Sweden thumped Azerbaijan 6-0 thanks to four goals from Viktor Gyökeres, and Slovakia defeated Estonia 1-0 to finish one-two in Group C1.
Minnows Malta and Andorra drew 0-0 in Group D2.
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