After a tepid opening gambit to the contest, Stoke almost took the lead through Ramadan Sobhi, but he could only turn his effort onto the crossbar after some memorable interplay between Joe Allen, Peter Crouch and Arnautovic to set up the chance.
Travelling Boro did little to strengthen their survival plight in the opening 45 minutes, although winger Adama Traore did his best to provide some sort of threat using his pace down the left.
The game was suddenly brought to life in the 29th minute when Arnautovic brought the ball out of the air, beat two defenders and then rounded Boro stopper Victor Valdes in order to give Stoke the lead.
Boro had not scored in their last three league games prior to meeting Stoke, and their bid to earn three points was further hampered when Arnautovic returned to grab his second of the match before half time, this time turning home from a well-worked corner to make it 2-0 at the interval.
The Potters began the second half in a confident manner and they might have found an early third when Arnautovic broke beyond his marker, although his cross was unable to find a completely-unmarked Sobhi inside the box.
Boro might have found a route back into the contest after 60 minutes when Ben Gibson poked in from a Grant Leadbitter free kick, only for the linesman to flag for offside to further intensify the visitors' frustrations.
With very little to play for in the final minutes, Valdes did his best to keep the score down as he produced a strong save to deny Ibrahim Afellay late on, as the hosts strolled home comfortably to secure all three points.