Schroder added three strikes to the one she scored in Stockholm last Saturday, which was enough for Hacken to take the aggregate lead back to Nordic Wellness Arena.
The home side's head coach, Elena Sadiku, kept faith with the starting XI that got the job done in the first leg, and the faith she put in those players was rewarded with two goals in the opening nine minutes that left Hammarby with a mountain to climb.
The first arrived in the sixth minute, when a free-kick launched from the haflway line was cleared, but Hacken kept the pressure on by moving the ball well on the left, from where Alva Selerud's square ball found Schroder on the edge of the box, and her turn and hit was just out of Melina Loeck's reach.
Getingarna did not rest on their laurels, and instead pushed for a second, which arrived three minutes later, when a clipped ball forward from Monica Jusu Bah was cushioned superbly by Anna Anvegard into the path of Schroder, who poked the ball past the goalkeeper with ease.
Hammarby boss William Stromberg had seen enough after only 14 minutes, bringing on Fanny Peterson for Gudrun Arnardottir to change the shape of his side. But just when a Schroder volley caught by Loeck was suggesting the substitution was having little impact, Bajen got themselves a goal.
From the next attack, the visitors worked the ball out to Stina Lennartsson on the right, and she dug out a cross that was headed home by Svea Rehnberg with 26 minutes on the clock.
The Stockholm side were then enjoying one of their best periods of the game, as Peterson came close to levelling on the day with a strike too close to goalkeeper Fanney Birkirsdottir, before Elin Sorum flashed a good effort just wide, but half-time arrived with Hammarby needing two goals to level the tie.
However, Hammarby were back within one just two minutes after the restart, when a deep Vilma Koivisto corner was met by Sorum, and her header found the top corner, out of Birkirsdottir's reach, to make it 2-2 in Gothenburg.
But credit to the home side, they refused to bow to pressure, and restored a two-goal aggregate lead just six minutes later. The impressive Schroder unlocked the Hammarby defence with a brilliant one-two with Anvegard to send her through on goal before burying the finish to complete an excellent hat-trick.
That goal knocked the stuffing out of Hammarby, who failed to register another shot on target until deep into added time when Vilde Hasund saw a header from a corner saved. Even had that gone in, it would likely have been too little, too late, as Hacken closed out the 3-2 win in front of their home fans and the celebrations could begin.
The 4-2 aggregate victory sees Hacken become the first Swedish club in men's or women's football to win a European trophy since Umea won the Women's Champions League in 2004. Hammarby must be content with being the bridesmaids again, having also finished as runners-up to Hacken in the Damallsvenskan last season, though they have another chance to get one over on their rivals when they meetin the Svenska Cupen final in two weeks' time.