York, of the Blue Square Premier and 99 places beneath Bolton in the league rankings, were backed by almost 5,000 travelling supporters and fashioned the better chances of the first half.
They continued to create openings in the second 45 minutes, but Coyle's input, together with late goals from England striker Davies and Swedish frontman Elmander, helped Bolton sneak through to the fourth round and Coyle revealed how he challenged his team following their tepid first-half display.
"I said, `You'd better get your act together'," he explained. "Because the way my team play we're not good enough at this stage of our development to think we can flick a switch and play at any given time.
"We have to play on the front foot with a real intensity for the whole duration of a match. Any less then we're always going to find difficult moments in a game, no matter who we play. We didn't have enough intensity in the first half.
"The introduction of Kevin Davies gave everyone a lift and when you're able to bring Matty (Taylor) and Johan into the fray you know there's real quality on the ball, and so it proved."