Facing Roger Federer on grass is usually a daunting enough prospect but lining up against the Swiss with a dodgy stomach could make Robin Soderling feel more than a little queasy on Monday.
World No 2 Djokovic, who received a first-round bye, broke his opponent's serve four times to extend his winning run to 19 matches.
"Just having your name on the trophy would be a dream come true," said the Australian.
Debutant Stefanos Tsitsipas crushed defending champion Alexander Zverev 6-3, 6-2 on Wednesday to book his place in the last four of the ATP Finals with a match to spare.
Former world number one Andy Murray is in line to play Grigor Dimitrov in the opening round of the inaugural ATP Cup, with Bulgaria qualifying among the final six teams in the nation-based tournament.
Djokovic had his personal best winning sequence of 30 consecutive sets ended as Cilic played almost flawlessly aggressive tennis to draw first blood in a magnificent duel.
The 2009 champion who will next face Roger Federer, saved two match points in the fourth set of what was arguably the best of the tournament so far.
Big-serving Milos Raonic fought back from a set down to upset Rafa Nadal 4-6, 7-6(10), 7-5 in a pulsating quarter-final at the BNP Paribas Open on Friday.
Defending champion Andy Murray eased into the Wimbledon quarter-finals for the 10th successive year with a 7-6(1), 6-4, 6-4 victory over Benoit Paire on Monday.
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