World number seven Djokovic was playing his first regular ATP Tour event since the Italian Open in May, having only appeared at Roland Garros, Wimbledon and the Laver Cup this summer.
Djokovic got the better of Andujar 6-0 6-3 in one hour and 28 minutes in only his fourth match since he claimed a 21st grand-slam title at SW19 in July.
The Serbian won the opening seven games of the contest before Andujar finally got off the mark following a marathon game that lasted 20 minutes.
"The first match started off in a perfect way for me," Djokovic said, as quoted by atptour.com.
"I won seven games in a row and we were fighting in that eighth game for 15 minutes, I think. It was one of the longest games I've ever played in my life and I've played many, many games in my life. But credit to Pablo for fighting and playing a great match as well."
British number six Liam Broady joined the Serbian in the last eight in Israel after an excellent 6-4 4-6 6-3 win over fifth seed Botic van de Zandschulp.
Diego Schwartzman, seeded third, also exited the tournament in Tel Aviv following a 6-3 2-6 7-6 (7) loss to Arthur Rinderknech.
At another ATP 250 event, the Korea Open, Cameron Norrie ended his three-match losing streak by beating wild card Kaichi Uchida 6-2 6-2.
Top seed and US Open runner-up Casper Ruud also made it through to the quarter-finals, but only following a back-and-forth battle with Nicolas Jarry, which the Norwegian won 6-2 3-6 6-3.
Jannik Sinner triumphed in the last-16 at the Sofia Open by a 6-3 6-4 scoreline against Nuno Borges.
Holger Rune followed the Italian into the final eight with a 6-7 (1) 6-4 6-3 victory over Lorenzo Sonego.