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 and 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.
