The midfielder struck to secure successive wins for Mark Hughes' resurgent side in a deserved 1-0 victory at Villa Park.
The Potters' last away Barclays Premier League win came in Sherwood's first game in charge at Villa in February.
Arnautovic had already seen a goal disallowed in the first half while Charlie Adam's audacious 60-yard lob almost had the visitors in front
But Arnautovic won it 10 minutes after the break as slapdash defending from the hosts cost them.
They failed to mount a response after going behind and remain winless in the Barclays Premier League since the opening day, and o nly Sunderland and Newcastle's implosions stopped Villa slipping to the bottom of the table
Jack Grealish was benched in the week that saw him pledge his international future to England, while Mame Diouf replaced the injured Xherdan Shaqiri.
But without Grealish Villa lacked any spark with Stoke, worryingly for Sherwood, comfortable.
An early sweeping move left Villa open at the back and Bojan spotted Diouf's run only for Micah Richards to produce a superb block when the striker pulled the trigger.
Stoke dictated the play but the game lacked real quality until Adam almost repeated his wonder goal against Chelsea after 18 minutes.
The midfielder scored from 60 yards at Stamford Bridge last season and nearly did it again when a frantic Brad Guzan tipped aside his effort from 10 yards inside Stoke's half.
The visitors were on top but struggled to find the crucial breakthrough against a limp Villa side who looked there for the taking a nd 10 minutes before the break the hosts survived a scare
Bojan was granted time and space to pick out Arnautovic who ghosted in between the defence to redirect the cross past Guzan - only to be ruled marginally offside.
Idrissa Gana fired widely wide with Villa's only serious effort of the first half and there were signs the natives were restless.
Grealish replaced the struggling Joleon Lescott at the break but he failed to be Villa's inspiration and they were behind after 55 minutes.
There was little pressure on Stoke as Adam and Glen Johnson combined before the defender threaded the ball through to Arnautovic who arrived unchecked in the box to fire in his second goal of the season, off the post.
Richards headed wide immediately after as Villa desperately searched for the leveller, while Jack Butland claimed Rudy Gestede's low drive after 66 minutes.
That it was the first time the goalkeeper had been seriously tested all game spoke volumes about Villa.
They again looked bereft of confidence and ideas going forward and blew what small chances came their way and Gana smacked a free-kick into the wall in stoppage time as any hope slipped away
Source : PA