Celtic dominated their part-time opponents but had to wait until six minutes from the end for the breakthrough, when the 21-year-old midfielder Callum McGregor scored with a deflected shot on his competitive debut.
The manager was pleased with the victory and the performance, which steadily became more threatening after a laboured first half. The third-round draw takes place on Friday. It is still an either/or, of course, but KR's prospects nosedived when the McGregor shot finally found a way through.
Celtic will play their first European tie at Murrayfield next Tuesday with a degree of comfort.
The surroundings in Reykjavik could not have been more low key - three sides of the KR-vollur ground resembled a public park - but there was no mistaking the grave importance of the night for Celtic. Dermot Desmond, the major shareholder, flew in to witness the start of his new manager's reign.
"I love to win, I am here to win," said Deila. "But I also need to win in a certain way as well. Today was a step forward in how we want to play and that's what I'm most happy about. If we win and play s*** then I'll get more irritated. But today if we'd been zero-zero I'd have been perfectly happy because I know we're going to score goals in the next leg as well.
"Of course there's relief to get that first win. We deserved more from this game, but I'm looking forward and I took a lot of energy from this. I liked what I saw of the team in the second half and I want more. I'm happy of course that we won the game, but I'm most happy about the performance. The first half was sleepy, for me, it was sideways passing and I hate too much sideways play. In the second half we started to penetrate more and got the chances."
McGregor has been with Celtic since the age of nine but had not previously played a competitive game and was last season loaned to Notts County, where he scored 14 goals in 40 appearances for the League 1 club. "It's about how you work in training and what you do in matches and so far Callum it has been fantastic," said Deila. "He has big talent, but I want him to be more direct, to be more forward and link up more with the full-back."
McGregor himself said going out on loan last season had transformed him. "I'm just delighted to get the win and to score the winning goal was unbelievable. I found out I was playing a few hours before the game and I had a bit of time to get my head around it. I was a bit nervous at the start but once I was out there, it was fine. I do feel ready for this. Going on loan to Notts County in League 1 was a big help for me . . . It was an eye-opener. I feel like I've come back a better, stronger player. I feel mentally ready for this challenge.
"I have scored a lot of goals but that was the biggest of my career. The club is striving to get Champions League football and that might be the goal that puts us through. It does feel like a fresh start. The manager gave me confidence and that was a big night for me."