Former world number one Murray ended a six-match losing streak in the opening round on Tuesday, but was edged out after almost three and a half hours on court and three tie-breaks, losing 7-6(6), 6-7(3), 7-6(4).
Mensik is enjoying a breakthrough season on the ATP Tour.
He came through qualifying before losing in five sets to Hubert Hurkacz at the Australian Open and defeated Alejandro Davidovich Fokina in the Doha first round.
The world number 116 served 12 aces past Murray, who missed two set points in the first tie-break.
"I'm just happy. It was a tough match today," said Mensik.
"Andy is an unbelievable player, I know that. When I was young, I watched him win Wimbledon two times, so it's unbelievable that he can still compete with the best players in the world."
Mensik secured the victory on his first match point, after earlier failing to serve for the match twice in the deciding set as Murray battled back from 5-2 and a double-break behind.
