The big-serving Croat took his aces tally to 63 for the tournament as he ended Johnson's Wimbledon run and underlined his own status as one of the main challengers to the established favourites.
Cilic, the 2014 U.S. Open champion, is one of only two people, alongside Stan Wawrinka, to have won a grand slam in the last seven years outside of the game's dominant quartet of Roger Federer, Rafa Nadal, Novak Djokovic and Andy Murray.
Having reached the final at Queen's in the traditional Wimbledon warmup, his grasscourt game is in good shape and he eased past Johnson with minimum fuss.
The Croat took the first set when he broke in the 10th game and claimed the second in a tiebreak
After both players struggled to hold serve at the start of the third, Cilic took control before wrapping up the contest when the American dumped a forehand return into the net.
Next up for Cilic is Roberto Bautista Agut, who reached the fourth round with a shock victory over Japanese ninth seed Kei Nishikori.
(Reporting by Toby Davis; editing by Clare Lovell)
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