Despite holding a first-leg 3-0 lead and going ahead on the night, the Hoops crashed out of the Champions League after a 5-1 defeat to Austrian champions LASK on Tuesday.
The home support again turned their ire towards the club's board for failing to invest vast bank reserves in reinforcing the squad before kick-off.
But a fast start quelled the unrest as Celtic made it nine points from their opening three Scottish Premiership matches.
Duran is one of a trio of attacking reinforcements the Glasgow giants have invested in ahead of the new season and the Colombian already has five goals in green and white.
The former Qarabag forward pounced on a howler by former Celtic goalkeeper Scott Bain to roll the opener into an empty net after just four minutes.
Duran doubled his tally and Celtic's lead before the half-hour mark when he swept in Kieran Tierney's cross.
But Martin O'Neill's men tired following their exertions of extra-time in midweek and allowed Falkirk back into the game in the second half.
A Colby Donovan own goal reduced the visitors' arrears 18 minutes from time and Celtic had to survive a nervy finale.
In a miserable week for the rest of Scottish clubs in Europe, Hearts were the only one of five sides to progress as they beat Rapid Vienna to reach the UEFA Conference League proper.
The Jambos, who missed out on a first league title in 66 years last season in the final minutes of the campaign, moved up to second in the table with a 2-1 win over St. Johnstone.
James Wilson's opener and an Enes Alic own goal put Hearts in command before Ruari Paton pulled a goal back.
Dundee United thrashed Kilmarnock 4-0 despite the absence of manager Jim Goodwin due to a health issue.
Rangers, who are already eight points adrift of Celtic, are aiming for their first league win of the season when they visit Aberdeen on Sunday.