Mitroglou scrambled in the decisive goal for the Portuguese champions from Luisao's header in the second half.
Dortmund's Pierre-Emerick Aubameyang hit a penalty straight at keeper Ederson after Ljubomir Fejsa handled.
The German side were the top scorers in the group stage, but failed to find a way through a stubborn home defence.
Dortmund will host the second leg on 8 March, with a place in the quarter-finals at stake.
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The Bundesliga side were expected to pose a greater threat to Benfica after scoring 21 goals in the group stage.
Gabon international Aubameyang had contributed four of those strikes but epitomised an out-of-sorts Dortmund, scooping a close-range chance over the bar and missing his penalty before being substituted after an hour.
Dortmund's form has faltered since December, with the German side winning just two of their last seven league matches.
Thomas Tuchel's side suffered a shock defeat by the Bundesliga's bottom club Darmstadt on Saturday, and the malaise continued as they were held at bay by a stubborn Benfica display.
The Portuguese club have never lost a last-16 tie in the Champions League - having reached that stage three times - and will have a lead to defend in Germany after scoring with their only shot on target.
Benfica coach Rui Vitoria: "We fought, gave our best, and victories are also made of that.
"Congratulations to Ederson too, a keeper who showed his quality once again, this time against a team that scored so many goals in the group stage."
Dortmund coach Thomas Tuchel: "Aubameyang is not fully fit after the Africa Cup of Nations and his body language did not suggest that he would do better after missing the penalty.
"We played an outstanding game and conceded a goal with their first shot in the second half.
"It is an extremely complicated result, but I am very proud of how we played tonight. Even though I am very disappointed, I am proud of how we presented ourselves."
Match ends, Benfica 1, Borussia Dortmund 0.
Second Half ends, Benfica 1, Borussia Dortmund 0.
Foul by Marcel Schmelzer (Borussia Dortmund).
Ederson Moraes (Benfica) wins a free kick in the defensive half.
Corner,  Borussia Dortmund. Conceded by Eduardo Salvio.
Marc Bartra (Borussia Dortmund) is shown the yellow card for a bad foul.
Foul by Marc Bartra (Borussia Dortmund).
Raúl Jiménez (Benfica) wins a free kick in the defensive half.
Attempt missed. Christian Pulisic (Borussia Dortmund) right footed shot from outside the box is just a bit too high following a corner.
Corner,  Borussia Dortmund. Conceded by Eliseu.
Julian Weigl (Borussia Dortmund) wins a free kick in the defensive half.
Foul by Filipe Augusto (Benfica).
Attempt missed. Marc Bartra (Borussia Dortmund) right footed shot from outside the box misses to the left. Assisted by Julian Weigl.
Christian Pulisic (Borussia Dortmund) is shown the yellow card for a bad foul.
Foul by Christian Pulisic (Borussia Dortmund).
Pizzi (Benfica) wins a free kick in the defensive half.
Corner,  Borussia Dortmund. Conceded by Ederson Moraes.
Attempt saved. Christian Pulisic (Borussia Dortmund) right footed shot from outside the box is saved in the bottom left corner.
Christian Pulisic (Borussia Dortmund) wins a free kick on the right wing.
Foul by Eliseu (Benfica).
Substitution, Borussia Dortmund. Gonzalo Castro replaces Raphael Guerreiro.
Substitution, Borussia Dortmund. Christian Pulisic replaces Marco Reus.
André Schürrle (Borussia Dortmund) wins a free kick in the defensive half.
Foul by Filipe Augusto (Benfica).
Substitution, Benfica. Raúl Jiménez replaces Kostas Mitroglou.
Marcel Schmelzer (Borussia Dortmund) is shown the yellow card for a bad foul.
Foul by Marcel Schmelzer (Borussia Dortmund).
Eduardo Salvio (Benfica) wins a free kick on the right wing.
Erik Durm (Borussia Dortmund) wins a free kick on the right wing.
Foul by Lubomir Fejsa (Benfica).
Hand ball by Erik Durm (Borussia Dortmund).
Attempt missed. Lukasz Piszczek (Borussia Dortmund) header from the centre of the box misses to the left. Assisted by Ousmane Dembélé with a cross following a corner.
Corner,  Borussia Dortmund. Conceded by Franco Cervi.
Corner,  Borussia Dortmund. Conceded by Eduardo Salvio.
Substitution, Benfica. Franco Cervi replaces Rafa.
Attempt missed. Marco Reus (Borussia Dortmund) left footed shot from the centre of the box misses to the right. Assisted by Raphael Guerreiro with a cross.
André Schürrle (Borussia Dortmund) wins a free kick on the right wing.
Foul by Victor Lindelöf (Benfica).
Lubomir Fejsa (Benfica) is shown the yellow card for a bad foul.
Ousmane Dembélé (Borussia Dortmund) wins a free kick on the right wing.
Kostas Mitroglou gave Benfica a one-goal advantage in the first leg of the Champions League last-16 tie with Borussia Dortmund.