Jasse Tuominen put the Finns ahead after 21 minutes before Pukki converted a penalty in the 64th and wrapped up the points with their third goal 15 minutes from time.
The win secured second spot in Group J for Finland with 18 points, six behind section winners Italy, who play away to Bosnia & Herzegovina later on Friday.
Finland players and fans back home could barely believe it after they ended a run 32 failed qualifying campaigns for either the World Cup or European Championship.
Hundreds rushed on to the Sonera Stadium pitch at full-time and manager Markku Kanerva, a former teacher, conducted his post-match interview with fireworks going off in the distance.
Pukki, who has scored nine goals in nine games for his country, said:
I'm speechless, this is sick, we made it.
Finland's Prime Minister Antti Rinne even tweeted his support: "An incredible performance, congratulations and thank you! Oh Finland is, oh Finland is finally in the men's soccer championship!"
Finland join Czech Republic, England, Beligum, France, Italy, Poland, Russia, Spain, Turkey and Ukraine has nations that have already qualified or the tournament next summer.