The hosts took the lead after 20 minutes when Ibrahimovic lashed his shot past Joe Hart at the second time of asking, netting the first goal in his country's new stadium.
© PA Photos
Sweden could have doubled this lead just after the half-hour mark when Mathias Ranegie found himself with the whole goal to aim at, but the striker somehow fired his shot over the bar.
Gerrard looked determined to get a goal to his name in his 100th appearance for England, but Andreas Isaksson was equal to his efforts in the second half.
Sweden levelled the scores in the 77th minute as Anders Svensson floated a lofted ball to Ibrahimovic, who chested it down before smashing a volley past Hart.
Ibrahimovic then completed his hattrick with just over five minutes on the clock, beating Hart with a low free kick from almost 30 yards.
The PSG striker put the icing on the cake for his side, and himself, in stoppage time as he took advantage of Hart being off his line to fire a bicycle kick in from all of 30 yards.