The Serbian second seed was broken in the third game when he double-faulted but hit back immediately to level at 2-2.
Djokovic, chasing a record-equalling eighth Wimbledon men's singles title, broke again in the eighth game before serving out for the set.
The covers were rushed on as soon as the defending champion had wrapped up the set, with rain falling in southwest London.
The roof was closed but there was a lengthy delay featuring multiple inspections by officials and the players.
Djokovic, who was clearly unhappy with the slipperiness of the surface, at one stage emerged with a towel, which he used to rub the surface of the court to laughter from the crowd.
Members of the ground staff then used hand-held machines to try to dry the surface in farcical scenes on the world-famous court.
The match eventually resumed after a delay of almost 90 minutes, with the Centre Court roof reopened.
Djokovic, seeking to match Margaret Court's all-time record of 24 Grand Slam singles titles, was unfazed by the delay, breaking in the first game of the second set to establish an iron grip on the match, repeating the feat to seal the set.
Cachin, ranked 68th, dug deep in the third set, with neither player able to force a break in the early evening sunshine.
Djokovic was immediately on top in the tie-break, surging into a 6-2 lead and he wrapped up the match when Cachin netted.
The 36-year-old Serbian is chasing the first calendar Grand Slam in men's tennis since 1969 after winning the Australian Open and French Open this year.
