Sean Raggett scored the only goal for the Vanarama National League side with a well-taken header in the 89th minute of the evenly contested tie at Turf Moor.
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While Dyche bemoaned a lack of clear-cut chances for his side, he held his hands up and credited Lincoln for a deserved win.
“I think they had once chance and that was the goal, so I am not too critical about the shape and the defensive side of the team.
“The technical detail in the team – and we have created enough to score a goal, and we did not do that – we have not kept the ball that well. The pitch is a bit tricky at the minute, we know that, unfortunately because of the weather up here.
“But no excuses, they approached it in the way cup sides should when you are the lower-ranked team. You know you are underdogs for a start, so you have that kind of feeling like: ‘What is the worst that can happen?’
“All credit to them, let’s make that clear, you have to find a way to win. No matter how football comes around, I am a great believer in finding a way to win. Whatever style you need, you have to find a way. And they found a way to win.”
He added: “The madness of football is that if we take our chance early, then it changes everything. But we did not and I believe in one of my mad laws of football, and that is the law of one more chance, and they had one and they took it.
“That is the marvel of the cup and unfortunately we have been on the wrong side of it today. I have had my good times in this competition, just not today. The fairy tale continues, it is just one of those things, and I wish them well beyond today, that is the way it goes.”