Poulsen struck 11 minutes from time with a precise, low shot after a cross by Nicklas Bendtner.
Denmark, who have 21 points from nine matches, will finish top of Group One ahead of Sweden, who have 15 points.
Portugal and Hungary, who were playing in Lisbon after a later kick-off, were vying with the Swedes for second place in the group and a playoff spot.
"It's great (to qualify)," Danish coach Morten Olsen said. "We have fought for it and played our best match so far in the qualification. It's a huge experience for all of us to qualify at home against Sweden."
Poulsen said: "It was amazing to score. This is the biggest moment for me as a player so far in my career. I'm happy and proud. This is a goal I have always dreamt of scoring."
Sweden were the more aggressive side at the start of the second half, with two big chances in the first two minutes.
Henrik Larsson found the net after 69 minutes when strike partner Zlatan Ibrahimovic threaded a pass through the defence into the area but he was ruled offside.
Ibrahimovic, who has been in top goal-scoring form for his new club Barcelona, had been seen as the big hope for Sweden but the Danes contained him.
"They have made it difficult for us now," said Ibrahimovic. "I still believe we can make it to South Africa."
Denmark have not lost a World Cup qualifier at home since 1981 and return to the World Cup finals after missing out last time.
