Finland and Sweden brought the number of qualified teams to 12, joining Spain, Italy, world champions France, England, Belgium, Russia, Ukraine, Poland, Turkey and the Czech Republic.
Eight more spots for the 24-team event – being hosted in 12 countries – are up for grabs in the group stage, which finishes on Tuesday, while the remaining four berths are determined in play-offs in March.
Finland became the 34th nation to make the finals in the biggest success of its 112-year football history.
Jasse Tuominen opened the scoring in the 21st minute in Helsinki for the completely dominating hosts after a shot from Pukki was blocked.
Pukki converted a penalty in the 64th and the Norwich striker struck again in the 75th to raise his campaign tally to nine goals.
The 11,000-crowd roared its approval throughout the match and then invaded the pitch in absolute delight after the final whistle, with thousands more celebrating all over the country.
“I have no words. Indescribable,” Pukki told broadcasters Viasat. “We have made history.”
Prime Minister Antti Rinne tweeted: “An incredible performance, Congratulations and thank you! Finland is finally in the men’s football championship!”
Meanwhile, Italy made it nine wins out of nine in the campaign – and 10 out of 10 for coach Roberto Mancini, with their win in Bosnia.
Defender Francesco Acerbi and Lorenzo Insigne were on target in the first half and Andrea Belotti wrapped up matters in the 52nd in Zenica.
“We knew it was not an easy game, Bosnia did now qualify, but they are a tough side. We, however, were soon on the right path,” Mancini said.
Ex-champions Greece won the other group game 1-0 in Armenia.
In Bucharest, Marcus Berg headed the opener in the 18th minute and then set up Quaison for the second in the 34th, which lifted them four points clear of Romania and Norway with one game to spare.
There were reports that Sweden’s Alexander Isak was the target of racist chants from the home fans.
Spain clinched the group win another five points ahead of the Swedes with a 7-0 demolition of Malta, which saw as many scorers.
In-form Alvaro Morata bundled the opening goal over the line from close range to score in seven consecutive games for club and country, and Santi Cazorla made it 2-0 before the break with his first national team goal in four years.
Pau Torres, Pablo Sarabia, substitute Dani Olmo and Gerard Moreno then struck for four goals in nine minutes between the 62nd and 71st minutes, before Jesus Navas completed the rout with five minutes remaining.
In Group D, Robert Skov and Tottenham’s Christian Eriksen scored a brace each, and Christian Gytkjaer and Martin Braithwaite also found the net against minnows Gibraltar.
The 1992 champions have 15 points – one more than Switzerland, who edged Georgia 1-0 from Cedric Itten’s 77th-minute header, and three more than Ireland who were idle.
But there is still all to play for on Monday (Tuesday AEDT) when Denmark visit Ireland and the Swiss are in Gibraltar.
Denmark can most likely not afford to lose, given that the Swiss are heavily favoured to win their fixture.