However, Celtic burst out of the traps eager to get revenge and three first-half goals from defender Mikael Lustig, striker Leigh Griffiths and winger Patrick Roberts had the tie finished before the break, with the second half a mere formality.
The Scottish champions will fly to Kazakhstan next week and while Astana, who booked their place in the next round with a last-gasp winner against Lithuanian side Zalgiris, will present a bigger challenge, Rodgers will be pleased with the way his side handled this potentially tricky match.
Celtic fans were given a pre-match boost when Bobby Lennox, who played in the club's European Cup final win over Inter Milan in 1967, and Rosemary Burns, wife of former Celtic player and manager Tommy Burns, helped unfurl the 2015-16 Premiership flag to raucous cheers.
Rodgers' attacking set up - Erik Sviatchenko was the only recognised centre-back - saw the visitors blitzed from the first whistle.
It looked like the home side had been awarded a penalty in the ninth minute when midfielder Stuart Armstrong went down in the box after a trip by defender Ryan Casciaro but, after consulting with his assistant, referee Bartosz Frankowski gave a free-kick at the edge of the box which Griffiths blazed over.
Midfielder Callum McGregor then cracked the bar with a deflected strike from 20 yards before he had a toe-poked effort from 12 yards pushed past the post by Imps keeper Raul Navas for another fruitless corner.
The breakthrough was, however, inevitable and Lustig's goal in the 23rd minute came when Lincoln failed to clear Armstrong's curling free-kick into the packed box.
Skipper Scott Brown was involved and when the ball fell to the Sweden full-back he turned and swept it into the net.
The Gibraltarians' resistance was shattered and three minutes later Griffiths drove in a second from the edge of the box after cutting in from the right.
In the 29th minute the impressive Roberts worked a one-two with McGregor and curled the ball past Navas from 10 yards for number three with the keeper denying Celtic striker Moussa Dembele six minutes later.
Lincoln would have welcomed the interval, but early in the second half striker Lee Casciaro, who scored in Gibraltar, almost notched again when he got behind Sviatchenko, only to screw his shot wide of the far post.
The visitors were enjoying some possession for the first time in the match but it was Griffiths who should have scored in the 65th minute when he met Lustig's pinpoint cross with his head, only to miss the target.
The luckless Dembele did the same with an angled drive from a more difficult position before he was replaced by Nadir Ciftci - Kristoffer Ajer and Emilio Izaguirre had already come on as substitutes - before Sviatchenko headed a McGregor corner past the post.
There was time for Navas to make a point-blank save from Griffiths, who then had the ball in the net only to be ruled offside, but by then it mattered little.
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