The match did not resume until the evening, when Djokovic promptly lost the first two points and the set, but he began the second well and in the end the slippery conditions hampered Fognini more, with one heavy fall at the start of the third leaving the Italian struggling.
Djokovic capitalised and wrapped up a 6-7 (7/9) 6-2 6-2 victory to set up a second-round clash with either Andy Roddick of the US or Slovakia's Martin Klizan.
Fifth seed Jo-Wilfried Tsonga was also in a spot of trouble against Brazil's Thomaz Bellucci.
Bellucci edged the opening set on a tie-break and the second and third sets were both close, but Tsonga, who has reached the semi-finals of Wimbledon for the last two years, came through 6-7 (5/7) 6-4 6-4.
Fourth seed David Ferrer of Spain had a much more straightforward time against Canada's Vasek Pospisil, winning 6-4 6-4, while Japanese 15th seed Kei Nishikori defeated Australia's Bernard Tomic 7-6 (7/4) 7-6 (7/4).
The courts became increasingly slippery as the rain fell and play was officially called off for the night just after 8pm with six men's matches completed.