The match had been carried over from yesterday after bad light stopped play, with Murray trailing by two sets to one but leading 4-2 in the fourth.
After levelling the match at two sets all, Murray had break point opportunities in the decider but couldn’t capitalise and at one stage was just two points away from crashing out of the tournament.
With no tie break in the final set, the Brit finally broke serve at 5-5 in the before serving out the match to secure a 3-6 3-6 6-0 6-3 7-5 win in 3 hours and 41 minutes.
Murray will now face French qualifier Mathias Bourgue in the second round tomorrow.
Murray levels, but Stepanek refuses to give in
After resuming a break ahead in the fourth, Murray spent little time in levelling up the match to force a final set decider.
The Brit subsequently had a break point in the opening game of the fifth, yet it was saved by Stepanek who was showing little signs throwing in the towel.
Just like in the opening two sets from yesterday, the Czech spent a large amount of time up on the baseline and was able to flatten out his shots to pick holes in the Murray defence.
The cold conditions once again allowed Stepanek time on the ball and opportunities to disrupt Murray, with a range of drop shots and sneak attacks to the net.
In the fifth game of the set Stepanek saved another break point with a well-executed and unanticipated serve-volley move.
His constant antics and positive hand gestures to the Philippe Chatrier crowd was a sign that he had rediscovered his mojo and momentum, which had clearly deserted him in the closing stages of play yesterday.
Get out of jail
At 5-4 up Stepanek almost conjured up a match point but hit the top of the net at 30-30 on Murray’s serve.
However, the tables turned and they turned quickly.
After holding on, Murray went on to break Stepanek’s serve in the very next game before serving out the match and booking his place in the second round.