The McDiarmid Park side took the lead in the first period in Austria through Chris Kane but had to settle for a draw when Mamadou Karamoko converted a second-half penalty following a silly challenge from Jamie McCart on the hour mark.
The Scottish side had a string of chances early in the second half and could have doubled their advantage when Glenn Middleton took a heavy touch when he looked certain to score from Michael O’Halloran’s break.
But Davidson believes it is a sign of how far his side have come that his players are disappointed to have drawn with a side that made the last 16 of the Europa League in 2019.
"They believe in themselves," he said. "They are all disappointed in the dressing room after conceding the penalty, and probably the last five minutes.
"But that sums them up, where their mindset is just now. They followed the game plan, they were brilliant, tried to play the right way.
"For St Johnstone to come away probably slightly disappointed with a draw, it shows the expectations and how we performed.
"I was absolutely delighted. If you asked me to go and play LASK with their European pedigree and come away with a draw and give ourselves a chance to qualify for the league, then I would have taken it.”
The Saintees lost 4-2 to Galatasaray in the Europa League third qualifying round after recording a 1-1 draw in Turkey, and Davidson is hopeful that a raucous home crowd can help spur his side on to the group stages of UEFA’s brand new competition.
"To qualify for this league is probably as big a game as St Johnstone have had in their history, so it's an enormous game,” he said of next week’s tie.
"The people of Perth missed out on two cup finals last season and to be fair this is a cup final for us on Thursday.
"We will have a great atmosphere again at McDiarmid, we got a full house last time."