After enjoying an early spell of possession, midfielder Lina Magull put the Germans in front in the 16th minute with an acrobatic volley through the legs of Swedish goalkeeper Hedvig Lindahl.
Sweden equalised six minutes later when winger Jakobsson latched onto a long ball straight down the middle from Linda Sembrant, and slotted home past goalkeeper Almuth Schult.
German coach Martina Voss-Tecklenburg threw on midfielder Dzsenifer Marozsan, who had not played since breaking a toe in their tournament opener against China, at the start of the second half but it was Sweden who took a shock lead.
Schult did well to turn away a powerful header by Fridolina Rolfo but the rebound fell kindly for striker Blackstenius, who fired home from point-blank range to get her second goal in as many matches.
Marozsan missed a glorious chance to force the game into extra time when she headed wide with the goal gaping in the 80th minute but Sweden held their nerve to reach their fourth World Cup semi-finals.
Sweden will now face Netherlands in the semi-finals, after the Dutch comprehensively defeated Italy 2-0 earlier in the day.
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