The match came to a stop as the spectator was attended to, with Djokovic providing him with a towel before he left his seat for treatment.
The world number two was nearly broken when play resumed but served out for the match to win 7-5 6-1.
Djokovic will play Bosnian Damir Dzumhur in the third round.
World number one Rafael Nadal and number three Roger Federer will start their campaigns at the tournament on Wednesday.
Nadal will play fellow Spaniard Fernando Verdasco, while Federer faces Milos Raonic, of Canada.