British number one Murray, 29, beat the Slovak world number 35 6-3 6-7 (5-7)6-0 to set up a tie with French world number 25 Gilles Simon.
The winner will meet American John Isner, ranked 27 in the world, in round three.
Scot Murray is looking to close the gap on world number one Novak Djokovic, who is not competing in Austria.
Were Murray to win the Erste Bank Open and the Paris Masters, which begins on 29 October, Serb Djokovic would need to reach the Paris final to retain top spot in the rankings.
World number two Murray is aiming to become the oldest player to reach number one for the first time in his career since Australian John Newcombe, who achieved the feat aged 30 on 3 June 1974.
Murray led 4-3 and 6-5 after breaking serve in the second set, but 27-year-old Klizan - playing his best tennis of the match - twice broke back to force a tie-break as Murray lost his composure.
Although Murray continued to shout and admonish himself, he dominated the final set, winning 27 of the 35 points played.
World number two Murray is aiming to become the oldest player to reach number one for the first time in his career since Australian John Newcombe, who achieved the feat aged 30 on 3 June 1974.
Murray led 4-3 and 6-5 after breaking serve in the second set, but 27-year-old Klizan - playing his best tennis of the match - twice broke back to force a tie-break as Murray lost his composure.
Although Murray continued to shout and admonish himself, he dominated the final set, winning 27 of the 35 points played.
