Dominic Thiem on Tuesday cruised into the semi-finals of the French Open defeating Alexander Zverev in straight sets.
Djokovic eased past Spanish veteran Verdasco 6-3, 6-4, 6-2 and will play in the last eight at Roland Garros for a record 12th time.
World number 38 Khachanov has now lost at the last-16 stage at Roland Garros twice in a row, while Zverev ended his wait for a maiden major last-eight appearance at the 12th attempt.
Zverev is looking to become the first German man to win the title since 1937
In all the four Slams, it's only the sixth time in the Open era that the defending champion has perished at the first hurdle.
Alexander Zverev has emerged as the main threat to Rafael Nadal's French Open crown in recent weeks as he bids to end Germany's 81-year wait for a men's singles title at Roland Garros.
Top seed Nadal raced away with the last five games to secure a 32nd Masters title and ensure a return to the world number one spot.
Alexander Zverev got his Rome title defence underway at the Italian Open on Wednesday with a straight-sets win over Italy's Matteo Berrettini as the German bids for back-to-back Masters triumphs after success in Madrid.
Alexander Zverev has long been marked out as a future star of the men's game and his victory suggests the German can be a major threat at the French Open later this month.
Rafael Nadal reached his 12th final in Monte Carlo earlier Saturday by seeing off Grigor Dimitrov 6-4, 6-1.
The top seed is bidding for an outright record 31st Masters title this week and extended his run of consecutive sets won on clay to 34.
Alexander Zverev came through a marathon final set under the lights to win 4-6, 6-2, 7-5 after two hours and 39 minutes.
The 12-time Grand Slam champion Novak Djokovic has still not reached a quarter-final since Wimbledon last July after struggling with an elbow injury.
Rafael Nadal returned to the top spot in the men's ATP world singles rankings, dethroning Roger Federer.
Rafael Nadal leads Spain into a home quarter-final against Germany at Valencia's iconic Plaza de Toros bullring.
John Isner came back from a set down to beat Alexander Zverev in two hours and 29 minutes.
World number five Zverev, 20, is the highest-ranked player left in the draw of the hardcourt tournament.
Maria Sharapova was sent crashing out of the Australian Open in a crushing defeat by Angelique Kerber Saturday as Simona Halep survived a titanic struggle to make the last 16.