Jonathan Walters scored the only goal in the first leg of Stoke's Europa League third-round qualifying tie against Hajduk Spilt.
The Republic of Ireland international nodded in from Matthew Etherington's cross after three minutes as the Potters made an impressive start.
Kenwyne Jones had a goal disallowed before Dean Whitehead and Robert Huth came close, but a dominant Stoke could not extend their lead.
The second leg takes place on 4 August.
Hajduk's visit was only Stoke's third home European game and the result far from guarantees Tony Pulis's side's progress to the play-off round.
After Walters' opener, Jones appeared to have added to the advantage a couple of minutes later, only to have the referee rule he had infringed in outmuscling the Hadjuk defence at the back post.
Centre-back Huth's decision to lurk in the penalty area after coming forward for a set-piece almost paid off, but his well-struck shot was deflected over the bar.
The German then headed wide from Etherington's corner after the break and Hadjuk goalkeeper Danijel Subasic had to be alert to palm away a cross from the on-rushing Walters as the hosts continued to press.
Former Tottenham defender Jonathan Woodgate remained on the bench as Pulis instead brought on centre-half Ryan Shotton as the hosts' frustrations increased in the closing stages.
And Stoke were forced to settle for a slender lead once Whitehead's drilled effort was well blocked in their last clear opening of the match.