The 29-year-old British star took his season record to 66 wins and just nine losses.
The 29-year-old Wimbledon and Olympic champion, who is closing in on Novak Djokovic's top ranking, took his season record to 66 wins and just nine losses.
Murray's best chance of taking the world number one spot will be to capture the title in Vienna, where he was champion in 2012, as well as next week's Paris Masters.
In the French capital Djokovic would have to reach the final to stay in pole position.
Australian Open and French Open champion Djokovic has spent a total of 222 weeks at number one in three spells -- including his current run of 121 consecutive weeks.
In his first meeting with Klizan, Murray had to save three break points in his first two service games before claiming the opening set.
Murray then served for the match at 6-5 in the second set, but world number 35 Klizan battled back to break and snatch the tiebreak.
Murray's greater range of shots proved key in the decider when he allowed Klizan just eight points.
The Slovak committed 58 unforced errors as his game plan of all-out attack came up short.
Murray will face Frenchman Gilles Simon in the next round with a 15-2 career record.
Big-hitting John Isner of the United States hit 52 winners and 31 aces in a 7-6 (7/1), 3-6, 7-6 (7/2) victory over 2004 champion Feliciano Lopez to reach the quarter-finals.
Ivo Karlovic of Croatia is also into the last-eight after 33 aces took him to a 7-6 (8/6), 4-6, 7-6 (7/5) win over Damir Dzumhur of Bosnia.