Two first-half goals from Hugo Almeida put the hosts on the front foot and, though Roei Kahat pulled one back four minutes after the break, Mehmet Aurelio replied two minutes later and further strikes from Egeman Korkmaz and Edu killed the game off.
It sent the Besiktas fans, who had been warned by their club to be on their best behaviour after reports of planned pitch invasions, home happy as they got the perfect start to their Group E campaign.
Relations have been severely strained between the two nations since a Turkish ship carrying aid to Gaza was sunk by Israeli forces last year, killing nine.
Reports suggested disruption, protests and a pitch invasion had been planned by home fans, leading the club to an issue an open letter to their supporters appealing for them to be hospitable towards their visitors.
Pro-Palestinian protesters made their feelings known before the match, while Turkish fans booed the Israeli side as they entered the field.
But the match got under way on time and the home support soon had reason to turn their boos to cheers.
Almeida latched onto a neat ball over the top from Mehmet Aurelio, before lifting it over Maccabi keeper Guy Haimov and into the net.
The visitors went in search of an equaliser but were restricted to shots from distance.
Shiran Yeyni saw his 20-yard effort stopped by Rustu Recber, before Kahat fired over and Rafi Dahan went wide.
When Nivaldo fouled Aurelio for a free-kick in a dangerous position, Besiktas came close to a second but Egeman Korkmaz headed Manuel Fernandes' set-piece wide.
Rustu was again called into action after 25 minutes, this time parrying away a 20-yard attempt from Yoav Ziv.
But despite the visitors' endeavour, it was Besiktas who got the second and again it went to Almeida.
Ricardo Quaresma sent in an inviting cross from the left and his Portugal team-mate did the rest, heading in the rebound after his initial attempt was saved by Haimov.
Haimov was in action again two minutes before the break when he pushed away a 20-yard free-kick from Manuel Fernandes after Savo Pavicevic had fouled Quaresma.
Maccabi stormed out of the blocks after the break and pulled one back when Kahat headed Elran Atar's cross past Rustu after a short corner.
Aurelio was the man marking Kahat on that occasion but he made amends moments later when he got his head to Fernandes' cross to restore the two-goal advantage.
And two minutes later it was 4-1 when Korkmaz nodded Quaresma's cross past Haimov.
Quaresma's 25-yard shot was held by Haimov and Tomas Sivok headed wide from close range as Besiktas dominated.
Rustu was called into action to parry Dahan's 20-yard hit, while Haimov did likewise to deny Edu, who then headed Quaresma's cross wide.
But the Brazilian go on the scoresheet two minutes from time, finishing from 12 yards to round off a thoroughly satisfactory night for the home side.