World number 13 Cilic looked to be heading for a routine victory when he claimed the opening two sets 6-3 6-2.
However, one break of Cilic's serve in the next two sets saw Brands take the match into a decider courtesy of 7-5 and 6-4 successes.
The fifth set looked to be heading for a tie-breaker as both players found it hard to make inroads on each others serve. Cilic struck in the very next game though to go 6-5 in front, before keeping his composure to book his place in the next round.
He will now take on Kei Nishikori for a spot in round four.