A Nacho Monreal strike and an Issa Diop own goal had helped the 'Gunners' overturn a one-goal deficit after Marko Arnautovic had handed the visitors a shock lead, but Welbeck was on hand to make the points safe in added time.
Hector Bellerin was able to get to the byline practically unchallenged down the right and he had the presence of mind to pick out Welbeck in the middle.
The England international showed impressive composure to spin and thunder an effort beyond Lukasz Fabianski in the West Ham goal form just seven yards.
With that strike Arsenal were home and hosed as they claimed the first points of the Unai Emery era.