Maya Yoshida was denied off the line while Che Adams air-mailed a free header in the opening half of play before Liverpool duly punished Southampton for their missed opportunities.
Sadio Mane scored for the first time against his former club and Roberto Firmino set the visitors up for what looked like a comprehensive win, only for goalkeeper Adrian to gift Danny Ings a late goal.
Ings appeared ready to fire Saints level, but couldn’t corral Jan Valery’s perfectly weighted cross as the chance went agonisingly past in the closing stages.
“The first half was not so bad,” Hasenhüttl acknowledged, with Saints coming off a 3-0 drubbing at Burnley to open the Premier League season. “We had a lot of chances and didn’t score, which is the only thing that wasn’t good.
“In the end, you see the difference from a player [Mane] that can score in that way. Not a lot of players can do that.”
Southampton stretched a sluggish Liverpool, which was jet-lagged from their triumph in the Super Cup in midweek, for the opening 30 minutes before Mane’s moment of brilliance nearing halftime.
Having looked like Saints would go a second consecutive game to start the campaign without a goal, Adrian’s mishap was a welcome stroke of luck with Ings pressuring the goalkeeper into playing the ball off his shin.
“We came back in the game to 1-2, and [Danny] Ings could have been the hero today if he scores that second goal,” said Hasenhüttl. “In the moment, there are just a few things missing.
“I think we were the better team in the first half. We were aggressive and created a few chances and that is the reason why I am positive overall.”
Southampton has the chance to build upon a positive performance when they travel to Brighton on Saturday.