They will enjoy a commanding lead ahead of the return in Gothenburg next week.
Putellas fired a rocket of a free kick into the top corner in the 11th minute and Pina doubled the home side's advantage in the 32nd minute with a well-placed finish.
Three minutes later Pina hit the crossbar and Putellas was on hand to tap in the rebound as the home side completely outclassed the Swedes in the first half.
Although the visitors did better after the break, Pina struck her second late on to complete the rout.
Earlier in the evening, Germany beat France 1-0 in Duesseldorf to take a slender lead into Tuesday's second leg, with the victorious semi-finalists set to contest the two-legged final on November 28 and December 2.
Klara Bühl scored the late goal in Dusseldorf after receiving the ball on the left in a well-worked counterattack, had plenty of space to rush to the edge of the box and fire a stunner from distance in the 79th minute to break the deadlock.
"The victory here feels incredibly good and also the way we played, we played football the way we wanted to. We can be incredibly proud of the team. Very, very strong match, I think the fans have noticed it," Bühl told broadcaster ARD.
"Of course we have the advantage, but we know we will have to work hard in France," the goalscorer said.
Germany welcomed back captain Giulia Gwinn, who suffered a knee injury in the opening match at Euro 2025 in July where the team reached the semi-finals, beating France on penalties in the quarter-finals.
"It was amazing, I had a lot of fun. I have to say that I was a bit emotional. It was a great game from us, the only thing we lacked was the efficiency (in front of the goal)," Gwinn said.
Germany coach Christian Wück added: "I was extremely annoyed that we didn't reward ourselves. The 1-0 was deserved, but not enough."
with DPA
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