Djokovic's team-mate Filip Krajinovic had teed up the chance for the world number two to send Serbia through from Group A with a 7-5 7-6 (7-5) triumph over Jo-Wilfried Tsonga.
After a group-stage exit at the ATP Finals in London last week, Djokovic made no mistake against Paire, winning 6-3 6-3 to secure a 2-1 victory for Serbia and send last year's runner-up France out in the process.
"The team is looking good, we won all six sets yesterday," Djokovic said at Caja Magica. "It was important getting a clean sheet against Japan coming into today’s match.
"Filip definitely gave me wings with his win, which was quite amazing. Tsonga was playing well and in good form and Filip was just fantastic on the court today. Credit to him."
Serbia, champion in 2010, will now face Russia in the quarter-finals and is joined in the last eight by Great Britain, which overcame Kazakhstan in a decisive doubles rubber.
Alexander Bublik's 5-7 6-4 6-1 win over Daniel Evans left the Group E contest tied at 1-1 heading into the decider.
However, Bublik, paired with Mikhail Kukushkin, failed to maintain his winning form, and Great Britain booked the remaining knockout place as Jamie Murray and Neal Skupski came out on top 6-1 6-4 in 72 minutes.
Group C winner Germany awaits Great Britain in the quarters, with Philipp Kohlschreiber's 6-4 6-3 defeat of Nicolas Jarry having sent it through at the expense of Chile.
Cristian Garin at least claimed a consolatory win – Chile's first of the tournament – with a superb performance against Jan-Lennard Struff, saving three match points en route to a 6-7 (3-7) 7-6 (9-7) 7-6 (10-8) success.
Glory for Kevin Krawietz and Andreas Mies in the doubles capped off a 2-1 triumph for Germany, with Garin's ultimately the only sets won by Chile at the Finals.