The former Barcelona forward tucked in his third goal of the season after 10 minutes, before miserly Stoke shut up shop to register a fourth straight clean sheet on the road.
Mark Hughes' side have lost just once on their travels, winning three and drawing three, and they moved to within a point of Saints after halting their six-match unbeaten streak in the Premier League.
Southampton boss Ronald Koeman said before kick-off that he did not expect a goal-fest between two evenly-matched sides - and he was right.
The solitary one that did come was well worked, though
Xherdan Shaqiri charged clear down the right, and as England full-back Ryan Bertrand ushered the Swiss winger inside, he played the ball across the 18-yard line to Marko Arnautovic
Austrian frontman Arnautovic helped the ball out to overlapping full-back Erik Pieters, whose cross was flicked in at the near post by the unmarked Bojan.
Shaqiri is starting to show the class which persuaded Hughes to shell out a club record £12million to snare him from Inter Milan in the summer.
However, he has yet to open his goal account and squandered two golden chances to break his duck.
Midway through the first half Shaqiri took Bojan's flick into the box superbly in his stride, but put his shot too close to the advancing Maarten Stekelenburg who was able to make a good block
Then moments after the interval Arnautovic reached the left-hand side of the Saints penalty area and pulled the ball back for Shaqiri, but this time he blazed his effort over the crossbar
Saints, by contrast, had offered little in attack, with Dusan Tadic off- target with a couple of shots and Virgil van Dijk heading Steven Davis' free-kick over the top
However, Shaqiri's second miss seemed to finally wake them up with Victor Wanyama firing narrowly wide from 20 yards
Southampton thought they should have had a penalty when Arnautovic sent Shane Long tumbling in the area but referee Lee Mason saw nothing wrong
Sadio Mane, who started on the bench due to a slight ankle injury, was thrown on with 20 minutes to go and forced Potters goalkeeper Jack Butland into a low save with a curling shot from outside the box
Arnautovic could have given Stoke some breathing space but he blazed over before fizzing a shot across goal and wide, while Ibrahim Afellay had a shot cleared off the line
However, Southampton's lack of punch in front of goal was summed up when Graziano Pelle sliced horribly wide late on
Source : PA
Source: PA