The Parkhead side laboured in the searing heat of the goalless first 45 minutes and were then stunned when Lee Casciaro, who had scored Gibraltar’s first international goal against Scotland at Hampden in 2015, sensationally put the part-timers ahead in the 47th minute.
Hoops striker Leigh Griffiths hit the woodwork twice as the Scottish champions battled for their dignity, but, buoyed by their stunning breakthrough, the home side held out for a result which ranks as the most embarrassing in Celtic’s European history.
The return game at Celtic Park next week should still be a formality, but Rodgers’ tenure has begun in the worst possible way.
It had promised to be a stroll for the Glasgow club.
There were around 300 Celtic supporters in the away end - some more interspersed with the home fans in the crowd of less than 2,000 - to see French striker Moussa Dembele make his competitive debut on his 20th birthday.
Attacking midfielder Ryan Christie was making his European debut, with last season’s top scorer Griffiths shunted wide on the right.
Swedish referee Andreas Ekberg punctuated the opening stages with his whistle, mainly for fouls against Celtic, although Hoops defender Efe Ambrose picked up a booking in the 10th minute for a foul on the night’s hero, Casciaro.
The Scottish champions, wearing their black and gold kit, huffed and puffed.
A couple of crosses from Griffiths failed to find a team-mate before Hoops goalkeeper Craig Gordon was forced to make a decent save from Liam Walker’s low 20-yard effort as the home side slowly began to edge forward.
In the 31st minute Dembele, who had a quite start to the match, headed in skipper Scott Brown’s free-kick, but was adjudged to have fouled keeper Raul Navas on the line.
Five minutes from the break Nava pushed a powerful drive from Dembele past the post for a corner which came to nothing before the Celtic striker fired an angled-drive wide of the far post.
The match exploded into life two minutes after the restart, following a Navas save from Griffiths.
Casciaro took advantage of Ambrose’s hesitation when chasing a long Walker pass to stab the ball past Gordon from 12 yards to widespread disbelief.
Moments later, as Celtic reeled in a state of shock, Lincoln midfielder Antonio Calderon fired just over the bar from 14 yards.
Rodgers brought on James Forrest and Stuart Armstrong for Christie and Tom Rogic before Griffiths’ shot from 10 yards was deflected on to the bar by Carlos Garcia before Lincoln captain Roy Chipolina completed the clearance for a fruitless corner.
As Celtic’s desperation grew, Armstrong blasted over the bar to groans from the increasingly worried Hoops supporters.
In the 72nd minute Nadir Ciftci replaced Dembele in what was Rodgers’ last throw of the dice, as scorer Casciaro came off to a hero’s reception to be replaced by George Cabrera.
Five minutes later, after Griffiths was fouled by Imps defender Joe Chipolina 25 yards from goal, the former Wolves striker cracked the bar with the free-kick, with Nir Bitton beaten to the rebound.
As Celtic stepped up the pressure in the final 10 minutes Navas saved from Ciftci, who subsequently missed the target with a header before the final whistle, after four added minutes, confirmed the Hoops’ embarrassment as the Lincoln players and supporters celebrated with justifiable pride.