The France Under-21 international replaced Danny Welbeck - absent with a knee problem - as the spearhead of Arsenal's attack and opened his account for the club in the second minute of this Group D clash at the Emirates Stadium on Wednesday.
A fine second-half strike from Alexis Sanchez - his 13th goal of an impressive season - sealed victory for Arsene Wenger's men, who came in for criticism after a 2-1 defeat to Manchester United in the Premier League on Saturday.
Arsenal needed just a point to be assured of qualification, but victory leaves them with a chance of sealing top spot in the group and avoiding a potentially tricky last-16 tie.
For Dortmund - already well off the pace behind leaders Bayern Munich in the Bundesliga - defeat ended their perfect record in the competition this season as a side missing influential playmaker Marco Reus through injury struggled to make much of an impact.
Both teams have seen their ranks depleted by injuries this season, and there was further concern for the hosts as first captain Mikel Arteta and then Sanogo were forced off during the second period.
That did little to disrupt their performance, however, as Wenger's men comfortably held on for their first win and clean sheet in four matches.
They were given an ideal start when Sanogo put them in front with the first action of note in the second minute.
The striker exchanged passes with Santi Cazorla - who came in following an ankle problem for Jack Wilshere - and, despite looking to be offside when the final ball was played, the Frenchman slotted a low finish through the legs of Roman Weidenfeller.
Sanogo could have doubled his tally soon afterwards, but he was ponderous in possession and saw his eventual shot blocked by Matthias Ginter.
Lukasz Piszczek flashed a dangerous cross across the face of goal in the 20th minute, but the delivery was just out of reach of a sliding Ciro Immobile.
Arsenal goalkeeper Emiliano Martinez, in the side due to Wojciech Szczesny's hip injury, produced a fine sprawling save to deny Henrik Mkhitaryan as Dortmund continued to carry a threat with half-time approaching.
Alexis, who had been relatively quiet in the first period, brought a smart save out of Weidenfeller early in the second half with a stinging low drive from just outside the area.
Alex Oxlade-Chamberlain almost marked his 100th appearance for Arsenal with a spectacular goal, but his dipping volley crashed back off the crossbar with Weidenfeller motionless.
The hosts did not have to wait long to double their lead, however, as Alexis cut in on his right foot and curled a delightful finish into the bottom right-hand corner in the 57th minute.
Oxlade-Chamberlain fizzed a shot narrowly over from the corner of the box midway through the second half, before Martinez denied Dortmund substitute Adrian Ramos with his legs in stoppage time as Arsenal sealed their place in the last 16 alongside their already qualified opponents.
