The Slovakians produced a lacklustre performance in poor conditions but an own goal by Polish defender Seweryn Gancarczyk in the third minute sent them through to next year's tournament in South Africa.
"We played well in these qualifiers, we were lucky today but we've also put a lot of heart into the game," Slovak coach Vladimir Weiss told reporters. "People deserved this and I'm glad that we made Slovaks happy."
Slovakia finished with 22 points in Group Three, two ahead of Slovenia.
Poland played an attacking game but were defied by a series of saves from Slovak keeper Jan Mucha.
The match was played in a near-empty stadium following heavy snowfall and the players struggled to keep their feet.
"The conditions are very hard, it is more a matter of accidents rather than (organised) plays," Poland goalkeeper Jerzy Dudek said.
Poland finished the qualifiers in fifth spot with 11 points.
Thank you for reading 5 articles this month* Join now for unlimited access
Enjoy your first month for just £1 / $1 / €1
*Read 5 free articles per month without a subscription
Join now for unlimited access
Try first month for just £1 / $1 / €1
