But the hosts did not heed that warning sign, and found themselves a goal down 90 seconds later.
Alexander Kerzhakov collected a ball into the left channel, and fired a dangerous low ball into the six-yard area where Kokorin nipped in ahead of Caldwell to get the deftest of touches through Boruc's legs.
With Celtic struggling to get to grips with the game, Dinamo continued to press and looked menacing - especially when Kherzakov received the ball, peeling out wide on the left.
Mowbray's men were finding it difficult to penetrate the Russians' tight-knit seven-man defensive structure, and in the opening quarter the best the hosts could muster was a close-range Fortune effort which was easily held by Vladimir Gabulov.
The Celtic midfield showed signs of rustiness, with Landry N'Guemo and Massimo Donati, as well as Fortune up front, guilty of surrendering possession too easily on occasion.
Chances did present themselves for Celtic, although the front pair of Fortune and McDonald lacked a predatory instinct.
Kokorin and Kerzakhov caused Celtic problems on the break
Fortune missed two gilt-edged chances to score within the space of 30 seconds.
First, the striker failed to connect properly with Aiden McGeady's cross at the back post, then, only three yards out, Fortune was unable to divert Shaun Maloney's excellent ball to the other post.
With 30 minutes played, McDonald showed good skill to cut inside a defender but should have done better than shoot at the goalkeeper from the corner of the six-yard box.
After ending the first period strongly, Mowbray's side pressed straight from the restart, and McDonald had a looping header cleared off the line after good work by Maloney - who was Celtic's brightest player on the night.
Kerzhakov gave the home side reminders of his threat, as twice on the counter-attack he used his strength and pace to trouble Caldwell and Glenn Loovens at the back.
Mowbray swapped his front pairing, bringing on Chris Killen and Samaras with half an hour to play.
The Greek striker had a fantastic chance to score with four minutes to go, but he headed straight into the thankful Gabulov's arms when unmarked from just a couple of yards out, and Dinamo take a valuable away win into the second leg.
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