The rain started 10 minutes before the scheduled race start, just after the drivers had observed the Monaco national anthem on the front of the grid and started to return to their cars.
Mechanics on the grid frantically rushed to change dry tyres to the intermediate wet in time for the 15:00 local time start, but the FIA delayed the formation lap to 15:09 and then 15:16.
Seven-time world champion Lewis Hamilton was seen shaking his head in the cockpit after being told of the delay, while reigning world champion Max Verstappen asked: "What are we waiting for? A dry track?"
The FIA mandated the formation lap would have to start under full wets.
The formation lap went ahead at 15:16 but the rain had intensified even more.
Ferrari's Charles Leclerc, on pole position, led the pack around for two laps behind the Safety Car before the race was red flagged at 15:20.
At one point, Leclerc said: "It's raining like crazy now."
The delay likely meant F1 missed an opportunity to start the race in the lighter rain and allow the cars to form a drier racing line on circuit.
An FIA statement said: "Race control was monitoring a severe downpour that was rapidly approaching the circuit, and as it arrived during the start procedure, the safety car start and its associated procedures were implemented. This was done for safety reasons in consideration that there has been no wet running this weekend."
At last year's Belgian Grand Prix, the FIA delayed the start of the race for three hours as they waited for a lull in a heavy rainstorm which never arrived. That race eventually finished without a single racing lap being completed, with half points being awarded, prompting a change in rules around points for similar scenarios going forward.