Left-back Nacho Monreal put Arsenal into an early lead with his fifth goal of the season, tapping in after home goalkeeper Aly Keita inexplicably let the ball slip from his grasp.
The visitors were also gifted the second goal in the 24th minute as Östersund tried to play their way out of their own area, with Henrikh Mkhitaryan latching on to the loose ball and seeing a low cross diverted into his own net by defender Sotirios Papagiannopoulos.
Östersund coach Graham Potter, a former defender for Stoke and West Brom in England, made two half-time changes in an attempt to change the flow of the game, but Arsenal all but ended the tie before the hour mark.
Mkhitaryan played in Mesut Özil, who jinked his way through and saw his shot burst through the hands of Keita and bobble into the net.
Östersund were gifted a chance to at least add a memorable goal to their fairytale run, but after Tesfaldet Tekie was brought down by Hector Bellerin, David Ospina saved Tom Pettersson’s injury-time penalty.
The comfortable win means that Arsenal manager Arsene Wenger will be able to rest his key players for the second leg at the Emirates in a week’s time.
“We did the job well, controlled the ball against a good side,” Wenger told BT Sport.
“Overall we played a serious game, so job done. They looked very nervous and we took advantage of it.”