Both teams had maximum points on the board after two rounds and they looked set to play out a goalless draw in a mostly low-key affair in Lisbon.
But Welbeck came up with the decisive goal in the 77th minute, rifling a fine finish through the legs of goalkeeper Renan Ribeiro to give Arsenal an 11th straight win in all competitions.
The Gunners move three points clear at the top of Group E, while Sporting are still well-placed to qualify for the knockout stages as they hold a three-point cushion in second spot.
Bruno Fernandes fired an early chance over the crossbar for the hosts before Henrikh Mkhitaryan's free-kick was beaten away by Ribeiro.
Nani lashed a drive over the angle before Sporting fans were angered when Sokratis Papastathopoulos appeared to drag Fredy Montero as he raced through on goal, referee Damir Skomina waving the appeals away.
Ribeiro twice denied Pierre-Emerick Aubameyang shortly after the restart with Arsenal upping the tempo.
Welbeck's celebrations after beating Ribeiro with a header were cut short in the 67th minute as official Skomina blew for a push on Bruno Gaspar.
However, he would not be denied again following Sebastian Coates' error, the England forward scoring his second goal in the competition this season to extend Arsenal's streak under three-time Europa League winner Unai Emery.
What does it mean? Welbeck winner justifies rotation
Emery shuffled his pack with Mesut Ozil, inspirational in Monday's Premier League win at home to Leicester City, left on the bench. Ozil's creative talents were missed as the Gunners found Sporting too tough to break down for much of the contest. Yet Arsenal have made a habit of winning matches in the second half of late and they did so again through Welbeck, backing up the Spaniard's selection policy.
Ribeiro shines in Sporting goal
Sporting are still trying to reach an agreement with Premier League side Wolves over their signing of Portugal goalkeeper Rui Patricio, but in Ribeiro they seem to have found a strong replacement. Romain Salin has been preferred in the Primeira Liga but a brace of smart stops from Aubameyang showed Ribeiro's abilities and he could do little about Welbeck's strike.
Coates mistake costs Sporting
A tight game always felt likely to be decided by a mistake and it was former Liverpool defender Coates who allowed Arsenal to maintain their momentum. He should have cleared the ball but instead slipped, with Welbeck making the most of the chance. Coates was also fortunate to see the match out as he appeared to clash with Stephan Lichtsteiner late in the game, with the officials not spotting the incident.
Key Opta Facts:
- Arsenal recorded their 11th successive win in all competitions; their best run of wins since October 2007 (12 in a row).
- This was Arsenal's first away win in Portugal after previously failing to come away victorious in six European visits there (D3, L3).
- The Gunners have scored 33 goals in all competitions this season - 23 have come in the second half of matches (70 per cent)
- This was the first time Arsenal did not concede a single shot on target in a European match since March 2015 (against Monaco in the Champions League).
- Unai Emery has suffered just one defeat in his 21 group stage games in the Europa League (W12, D8, L1, while at Valencia, Sevilla and Arsenal).
- Welbeck has been directly involved in five goals in his last five starts for Arsenal in all competitions (four goals, one assist).
11 - Arsenal have recorded their 11th successive win in all competitions; their best run of wins since October 2007 (12 in a row). Fresh. pic.twitter.com/KxDbVJaA02 — OptaJoe (@OptaJoe) October 25, 2018
Another away game for in-form Arsenal at the weekend as they make the short trip to Crystal Palace in the Premier League on Sunday, with Sporting hosting Boavista the same day. The two sides will face off again at Emirates Stadium when the Europa League returns on November 25.