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Sanogo turned provider shortly after the half-hour mark when he set up Jon Toral on the edge of the area, only for his effort to clip the outside of the post.
Arsenal continued to up the pressure before the interval and once more Sanogo was involved, this time finding space inside the Boreham Wood box, before getting a shot away which was hit straight at the keeper.
Abu Diaby's powerfully struck effort provided the Gunners' best opportunity in the first 15 minutes of the second half, but Russell was on hand once more to make the save, and he was needed again just one minute later to deny Benik Afobe's long-distance shot.
Arsene Wenger's side did eventually make the breakthrough, however, and it was Kristoffer Olsson who found the net with a clipped effort past Russell following Tomas Rosicky's through-ball to make it 1-0 after 69 minutes.
Chuba Akpom almost doubled the Premier League side's advantage with a little over 10 minutes remaining following a mazy run into the box, but his effort was well blocked.
Afobe made sure of the victory when he was brought down inside the area with four minutes remaining, getting back up to successfully convert from the spot to secure a 2-0 win for the Gunners.