Welbeck's 77th-minute effort has moved Arsenal onto nine points in Group E, and it means that the Gunners have now won their last 11 matches in all competitions.
The run is their best for 11 years, and it is also the first time that the English giants have beaten a Portuguese side away from home.
It took until the 16th minute for either side to have a shot, but Bruno Fernandes's effort from outside the Arsenal box was never threatening to find the back of the net.
Easily the best chance of the first period fell to Nani in the 31st minute following a mistake in the Arsenal defence, and the former Manchester United attacker's powerful effort just missed the top corner.
Arsenal were fortunate to keep all 11 players on the field late in the first half, meanwhile, when Sokratis Papastathopoulos pulled Fredy Montero to the ground. The referee did not regard it to be a foul, but if it had been, Arsenal's centre-back would have surely been sent off due to where the challenge took place.
Arsenal had 57% of the ball in the opening 45 minutes, but it was a first period that lacked genuine quality, and neither team did enough to deserve to lead at the interval.
Montero headed a Nani cross wide of the post early in the second half as Sporting looked to make the breakthrough, before Ribeiro had to keep out a Pierre-Emerick Aubameyang effort down the other end.
Welbeck had a golden chance to send Arsenal ahead in the 54th minute after meeting a low cross from Matteo Guendouzi, but the England international's effort was deflected wide of the post.
Guendouzi then tested Ribeiro in the 63rd minute as Arsenal searched for the opening goal of the match.
Welbeck had the ball in the back of the Sporting net in the 68th minute when he headed past Ribeiro, but the referee had blown for a foul, which cut the away side's celebrations short.
Welbeck was not to be denied in the 77th minute, however, as the Arsenal forward raced clear of the Sporting defence before sending a strike past Ribeiro to make the breakthrough in Lisbon.
The goal was no more than Arsenal deserved following their improvement in the second period, and there was to be no late response from Sporting, who lacked quality and invention on the night.
SPORTING LISBON (4-3-3): Ribeiro; Ristovski (Gaspar 46'), Coates, Andre Pinto, Acuna; Petrovic, Gudelj (Cabral 71'), Battaglia; Bruno Fernandes, Montero, Nani (Diaby 86')
ARSENAL (4-2-3-1): Leno; Lichtsteiner, Sokratis, Holding, Xhaka; Guendouzi, Elneny (Torreira 58'); Ramsey, Mkhitaryan, Welbeck (Lacazette 81'); Aubameyang (Iwobi 86')