The Potters, who went into the third round second leg with a 1-0 lead from last week's first leg at the Britannia Stadium, put in an impressively controlled display over 90 minutes, Pulis' players seemingly untroubled by the vociferous home supporters at Stadion Poljud.
Potters boss Tony Pulis said: "I'm very proud of the players and for the football club.
"The chairman (Peter Coates) has his family out here today. They will be absolutely delighted and I am very pleased for them.
"We were quietly confident. The important thing was to make sure that we came here with the lads full of confidence.
"We know Hajduk are a very famous and great club, and they have fantastic supporters - we knew the atmosphere would be brilliant.
"But we thought that if we played our game, and we played as well as we can, with the lads putting the effort and commitment in, then we'd have as good a chance as them of winning the game.
"This is the first time the club has played a European tie and actually qualified or gone through, so it's another new bridge we have crossed.
"We got into the Premier League a few years ago, then we got to Wembley (for the FA Cup final last season) and now we have crossed this bridge - they are three fantastic achievements in such a small space of time."