The Potters, playing in major European competition for the first time in 37 years, took the lead at The Britannia Stadium in the third minute when Walters was allowed space to nod in Matthew Etherington’s measured cross.
Despite dominating for much of the match thereafter, though, they were unable to find the second goal that would make next week’s trip to Croatia for the second leg more comfortable, as tiredness began to tell on Tony Pulis’ players.
Despite being relative novices in Europe compared to their opponents, it took no time for Stoke to find their rhythm against Hajduk and they were soon in front.
Etherington collected Rory Delap’s pass on the left flank and delivered a delightful cross on to the head of Walters, who calmly dispatched the ball into the corner of the net.
Moments later the Potters thought they had made it 2-0, but Kenwyne Jones’ low effort was ruled offside.
Walters knocked the ball down for Jones just after the quarter-hour mark, although the Trinidad and Tobago international could not keep his shot under control.
Ante Vukusic then tried his luck for the visitors, only to lash his strike wide.
Hajduk enjoyed a brief spell of pressure and the ball almost fell for Vukusic in front of goal before squirming through to Asmir Begovic.
But Stoke were soon back on the offensive, with Robert Huth and Jones both seeing shots blocked by some last-ditch defending.
Jones rose on the edge of the six-yard box to meet Jermaine Pennant’s free-kick but Hadjuk goalkeeper Danijel Subasic got down to save the striker’s header.
Subasic was then called into action again to collect Etherington’s drive.
The away side appeared to be struggling with balls into the box and were forced to make a goal-line clearance six minutes before the break when Ryan Shawcross flicked on Delap’s long throw.
The pattern of the game showed little sign of changing as the second half got under way, with Etherington sending in a dangerous cross which just evaded Walters and a corner that Huth failed to guide on target when he might have done better.
Shawcross then got his head on another Delap throw into the area which Hajduk managed to scramble behind.
Marin Tomasov unleashed an effort from outside the box for the visitors but Begovic dealt with it comfortably.
Stoke’s intensity appeared to have dropped a little with 15 minutes to go and Pulis brought on Dean Whitehead for Pennant.
Soon after, Subasic dived to push an Etherington cross away from the in-rushing Walters, before Whitehead saw a drilled effort blocked.
Etherington was then replaced by Ryan Shotton and the weary-looking hosts were unable to create much more in attack until Huth scooped an effort wide right at the death.