Reigning champion Djokovic, who is seeking a sixth triumph in Rome, struggled to get going against Tsitsipas and trails 6-4 2-1 ahead of the scheduled resumption of play on Sunday (AEST).
The No.1 seed lost serve in the first game and continued to make some uncharacteristic errors that saw him trail 4-1 at one point.
While Djokovic did earn a first break in the sixth game, Tsitsipas stood firm to take the opening set and was in total control when again breaking his opponent early in the next set.
However, soon after the Greek had opened up a 2-1 advantage, the worsening conditions led to a pause in proceedings and organisers were eventually forced to bring the day's play to an end. Djokovic, who converted just one of his six break point opportunities, will return to the court on Saturday and will need to improve if he is to join Rafael Nadal in the final four.