Kolesnik turned aside 33 of 35 shots but it was not enough to stop a Czech Republic that are undefeated through five games and moved five points up on Russia, which have a game in hand.
Tomas Plekanec scored early in the first and third periods for the Czech Republic but Kazakhstan gave themselves some hope when Nigel Dawes scored with less than five minutes to play in regulation.
But with Kolesnik pulled from the goal in favour of an extra attacker, Robert Kousal sealed the win when he scored into the empty net with 69 seconds left.
"It was tough for us today because we played yesterday as well," the Czech Republic's Jan Kolar told the International Ice Hockey Federation's (IIHF) website.
"We weren't too worried that it was taking a long time to get the game wrapped up. In a tournament like this you sometimes just have to grind out the win without playing pretty."
Up next for the Czech Republic is a game against Denmark on Sunday while Kazakhstan face Latvia on Saturday.
The United States, powered by a pair of goals from Nick Foligno, breezed by winless Hungary 5-1 in Group B action for their third win in five games to move into third place, two points ahead of Germany.
"In the first period we dominated," U.S. forward Auston Matthews said according to the IIHF's website.
"We had a ton of shots and we limited them to one. We were producing a lot of scoring chances, but their goalie played extremely well. In the end, we kind of broke him down a little bit and were able to get a couple on him."
Germany moved into contention for a quarter-final spot after beating Belarus 5-2, while Nicklas Jensen scored the overtime winner for Denmark in a 3-2 victory over Latvia.
(Editing by Frank Pingue)
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