Jon Walters' third-minute header at the Britannia Stadium means the Potters hold a narrow advantage over their Croatian opponents after the first leg of their Europa League third qualifying round tie.
Although they began to tire towards the end, Stoke dominated for much of the game and, particularly in the first half, had several chances to extend their lead.
With more than a fortnight still to go before the Premier League gets under way, it was an encouraging display from Tony Pulis' team and according to Delap, just reward for their graft over the last few weeks.
"The pre-season has been hard but it showed it's been worth it," Delap said.
"It's the first time I've played in July so credit to the manager and his staff that we were ready for it in such a short space of time, although every one of the lads came back fit."
The match was not only Stoke's first of the season but also their first in major European competition for 37 years.
The Potters faithful celebrated the occasion by roaring the team on throughout and Delap is determined to emerge from next Thursday's second leg at the Stadion Poljud with a place in the next round secured.
"I thought our fans were tremendous and their fans also created a real atmosphere," Delap said.
"I thought we played well and were a bit unlucky to come away with only a 1-0 win.
"I don't know who are the favourites. We are in the lead at the moment and it's half way.
"They will be looking to win the game at home but if we play the way we did here we should come away with the win.
"We want to do well in everything we're in, in the league and every cup.
"This is a chance we're not going to get too often if we're honest, so we have to make the most."
Delap proved a constant thorn in Hajduk's side with his trademark long throws, one of which led to a goal-line clearance to scramble away Ryan Shawcross' flicked header
The 35-year-old's famous weapon did not impress Hajduk coach Krasimir Balakov, who said: "I don't think it's right football. But we had to get used to it.
"We don't have that style but we had to adjust to the situation and we will have to do so again in Split."
Told of Balakov's criticism, Pulis said: "I don't give a hoot, like I don't with anyone else."
The second leg promises to be a testing assignment for the Potters, but Pulis simply wants his players to enjoy themselves.
He said: "We've never experienced it so we'll just go there and enjoy it, and if it comes off for us, brilliant. We'll give it a go."