The Scot has lost to a number of the world’s top 10 since the US Open – Hurkacz being one last month in Metz – but after a good start in Vienna he ended up becoming embroiled in a two-hour-40-minute battle with the Polish fifth seed.
Murray blamed a poor attitude for not getting beyond the last 16 in the European Open in Antwerp last week but there was no questioning his commitment on his return to the court.
The first eight games went with serve before the former British number one broke to allow him to take the set 6-4.
With both players holding their serve in the second it went to a tie-break, which Hurkacz won 8-6 thanks to two unforced errors.
Murray enjoyed two service breaks in the deciding set, saving two break points in the ninth game to win 6-4 6-7 (6) 6-3.
His opponent in the next round will be 18-year-old Spaniard Carlos Alcaraz, who beat British number two Dan Evans earlier in the day.
Murray is still on the comeback trail but he feels he is making improvements.
“My movement has been getting a little better with each match,” he said.
“A lot of my movement is about anticipation and when you’re not playing many matches – like I haven’t been in recent years – you don’t read the play quite as well.
“Now I’m starting to see the ball a little earlier and starting to react a little quicker, which means I will start to chase more balls down.
“It was a good match that could have gone either way. It was a good win in tough circumstances.”