The German then went on to hold the rest of his service games quite comfortably and with help of some big serves, closed out the match easily to reach the last 16, keeping his London bid well on course.
Rising star Zverev was similarly emphatic as he brushed aside Australia's new number one, teenager Alex de Minaur, 6-1, 6-4.
Djokovic saved both break points he faced on serve in the match and successfully broke Cecchinato's serve on four of 11 offerings. Del Potro had the knee wrapped but called it quits after losing the first set 7-5.
The Swiss looks some way short of his best form and for the second match in a row dropped a set.
Djokovic will play South Africa's seventh seed Kevin Anderson.
There wasn't much Gasquet could have done as Del Potro just brought his A game when it mattered the most.
Basilashvili's victory over an under-the-weather del Potro at the weekend was just his second ATP title, the first coming in Hamburg in July.
He will next face Borna Coric, who won his R2 match yesterday against Bradley Klahn easily.
Top seed and holder Federer starts his campaign on Wednesday against Russia's Daniil Medvedev.
The pair traded breaks at the start of the second, with Bautista Agut responding with a break to love courtesy of a rasping backhand return victor after a demoralising opening game of the set.
But at 4-4, the Swiss moved up a gear and played a brilliant, attacking return game to get the break at a crucial juncture and closed out the opener 6-4.
Bautista Agut had never even taken a set off Federer in their previous seven meetings but forced a decider only to be crucially broken at 4-4, and the Swiss served out to love to clinch the win. "I played every single point as if there was no tomorrow". Reflecting on 2017, he added, "Last year was a fairy-tale from start to finish, basically". The intense battle continued in the 2nd, but it was Del Potro who succeeded again, this time at 7-6 to wrap-up the match in an hour and 53 minutes.
Kristina Kucova, who caused an upset Monday after she knocked out Jelena Ostapenko, glided through her match with Viktoriya Tomova 6-1, 6-0.
The number one seed and defending champion was pushed all the way by the surprise victor of the Japan Open, but broke in the ninth game of the deciding set before serving out to reach the last 16.