World number two Murray, who beat Novak Djokovic to win the Italian Open in his last match, lost the first two sets to the 37-year-old Czech qualifier.
The Briton won the third set and was leading 4-2 in the fourth when the duo were called off at 20:22 BST, with Stepanek leading 6-3 6-3 0-6 2-4.
The match resumes at 11:30 on Tuesday.
Murray has not lost in the first round of a Grand Slam tournament since the 2008 Australian Open
Murray has recovered to win from two sets down eight times in his career.
The 29-year-old lost his serve in third game of the first set and was broken again in the ninth as Stepanek, the oldest player in the draw, drew upon his vast experience.
The second seed broke to love to start the second set but then immediately lost his own serve and Stepanek, ranked 127 in the world, broke again at 4-3 to carve out a two-set lead.
Murray, though, showed his resolve and raced through the third set in just 18 minutes as former world number eight Stepanek began to lose his way in the fading light.
Stepanek, a Wimbledon quarter-finalist in 2006, seemed to be stalling for time and received a warning after taking a toilet break and then changing his shirt.
It did not disrupt Murray's rhythm and the Scot extended his run of games to nine before the Czech held his serve.
The British number one was next to serve and seemed well placed to force a decider when play was called off for the evening.
Murray is bidding to join fellow Britons Kyle Edmund and Heather Watson in the second round, while compatriots Johanna Konta, Laura Robson and Aljaz Bedene are due to start their campaigns on Tuesday.
