Former champion Cilic, who rallied back from a two-set deficit to clinch a four-hour, third round match that ended after 2:00 a.m. local time, made a sluggish start against the 10th-seeded Goffin but had enough gas in the tank to advance.
Goffin, who was bidding to become the first Belgian man to reach the U.S. Open quarter-finals, enjoyed a nice start with a break of serve and grabbed a 5-3 edge before moving within two points of taking the first set.
But Cilic suddenly rose to the occasion and deployed his ballistic forehand to get back on serve at 5-5 before securing the opener after an error-filled tiebreak by both players.
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From there, the Australian Open runnerup settled into a groove and pulled away from Goffin, who was limited by a sore shoulder and unable to take advantage of an uneven serving display from Cilic.
"I started a little bit slower and he was playing good in the beginning and when he was serving in the first set at 5-4 it was a crucial break for me to get back in the game," said Cilic, who made just 46 percent of his first serves.
"After the first set it felt like he slowed down his first serve and at the end he told he's got some problems with his shoulder so it was a little but unfortunate for him and he was not playing the best the last two sets but I kept my rhythm and played well."
Cilic broke Goffin in the fifth game of the third set to pull ahead and then held serve the rest of the way before sealing the deal on his third match point with a crosscourt forehand.
The win puts Cilic into the quarter-finals of the year's final Grand Slam for the fifth time.
Next up for Cilic is a 2014 U.S. Open final rematch with Japan's Kei Nishikori, the 21st seed, who beat German Philipp Kohlschreiber 6-3 6-2 7-5 earlier on Monday.
(Reporting by Frank Pingue; Editing by Ian Ransom)
