Welbeck broke through on goal after Sebastian Coates’ error and the 28-year-old finished through the legs of Renan Ribeiro to hand the Gunners all three points in Lisbon.
Sporting certainly started the better of the two sides, with Bruno Fernandes twice shooting over from the edge of the box.
Former Manchester United winger Nani came close to giving the hosts the lead, with his dipping shot flying just inches over the bar.
The moment of controversy came when Fredy Montero was put through on goal before appearing to be pulled down by Sokratis Papastathopoulos.
Referee Damir Skomina waved play on after seemingly failing to spot the incident and the Greek centre-back was fortunate to escape without punishment.
Things picked up in the second half for the Gunners and Pierre-Emerick Aubameyang had two golden opportunities to take the lead but on both occasions was denied by the impressive goalkeeper Renan Ribeiro.
Danny Welbeck then thought he had broken the deadlock, sweeping towards goal after being found by Matteo Guendouzi but his close-range effort was blocked brilliantly by Andre Pinto.
Arsenal continued to press forward and Welbeck once again thought he had got himself on the scoresheet, nodding home from a Henrikh Mkhitaryan cross, but the referee had correctly blown up for a foul.
The England international did have the last laugh though, reacting first to Coates’ error in his own half and found the net to all but clinch Arsenal’s place in the next round.