Seventh seed Murray lost the first set in a match that swung back and forth, but neither player was able to find a way to victory on Court Suzanne Lenglen, meaning the pair will resume battle on the Roland Garros clay on Sunday with the score at 3-6 6-3 6-3 4-6 7-7.
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Both players had problems on their serve, with 17 breaks in the match before play was halted, while Murray was particularly vulnerable on second serve, the Scot claiming just 47 percent of second-service points.
The Scot made a bright start to the opener, a series of fine backhands seeing him earn the first break of the match and a 2-1 lead.
However, Kohlschreiber attacked on the second serve with plenty of success, and he rattled the Wimbledon champion by reeling off four straight games to take clinch the first set.
Murray was still well short of his best in the second, but managed to once more take an early break and soon found himself 3-0 ahead.
Kohlschreiber was able to earn the break back, but Murray instantly returned the favour as both players struggled to gain momentum.
The rest of the set went with serve as Murray levelled the match, and he opened up the opportunity to advance to round four when he clinically broke twice in the third to take a 2-1 overall advantage.
There was further positivity when he again made a quick start by winning the first three games in the fourth set, and although Kohlschreiber cancelled that break, Murray immediately responded to lead 4-2.
However, the two-time grand slam champion then capitulated, Kohlschreiber claiming four straight games, holding to love to send the match to a decider.
The fifth set proved to be as tense as the rest of the match, with Murray once again breaking at the first opportunity only to hand the advantage straight back before the pair both gave up their own serve again.
But with the score locked at 7-7 and with the light fading in Paris, the players were called off and will now play to a finish on Sunday.