The clash marks the return of the FA Cup runners-up to European competition 37 years after they last competed in the UEFA Cup.
On that occasion they were knocked out in the first round on away goals by Ajax, and the Potters have again drawn one of Europe's bigger names.
Manager Pulis told the club's official website: "It's a fantastic draw for us and an occasion that every single one of us are looking forward to.
"Hajduk Split are a hugely renowned football club with great tradition in European competitions, and it's also a lovely city for our supporters to go out and visit.
"The draw has only just been made but already my phone hasn't stopped ringing - you can sense the excitement building.
"It would be absolutely fantastic to get a full house at the Britannia Stadium and get the place rocking as we host European football here for the first ever time."
While Stoke will hope to win a European tie for the first time at the third attempt, Hajduk possess an impressive European pedigree.
They have reached the quarter-finals of the European Cup three times, most recently in 1994-95, while they were semi-finalists in the UEFA Cup in 1983-84.
Hajduk are also six-time winners of their domestic league, although for the last three years they have had to settle for the runners-up spot behind Dinamo Zagreb.
The first leg will take place at the Britannia Stadium on July 28 before Stoke travel to Split's Stadion Poljud on August 4.