Croatia, the 2018 World Cup runners-up, jumped above Russia to top the standings on 23 points from 10 games, one more than the second-placed Russians who will head into the March playoffs.
Croatia dominated a match played in driving rain but a string of good saves by visiting goalkeeper Matvei Safonov and a waterlogged pitch kept them at bay before Kudryashov's clumsy 80th-minute touch delighted the home fans.
The home team's coach Zlatko Dalic, who broke down in tears after the game as he came under pressure from fans and media during Croatia's patchy qualifying campaign, praised his players and the supporters.
"It's a relief, it was a titanic tussle and I want to congratulate all the players who took part in these qualifiers," Dalic told a news conference.
"A big thanks to the fans too, from the bottom of my heart. You're our pride and energy. Hence the tears, I don't cry very often but this victory was one of the greatest sporting moments of my life.
Needing a draw to qualify, the Russians defended with everyone behind the ball and soaked up intense pressure as the Croatians besieged their goal but failed to create clear-cut chances.
Andrej Kramaric headed straight at Safonov in the ninth minute before Marcelo Brozovic unleashed two long-range efforts, one sailing inches wide and the other drawing an acrobatic save from the visiting keeper.
Safonov kept out a close-range Mario Pasalic header with a superb stop in the 47th minute and the Russians appeared to be home and dry before they were undone by Kudryashov's error.
With waves of Croatian attacks losing momentum on a difficult pitch, left back Borna Sosa floated in a hopeful cross from the left and Kudryashov bundled the ball into his own net after it bounced awkwardly in front of him.
Russia missed their only chance to salvage the game from their first fluent move in stoppage time, Vyacheslav Karavaev firing over the bar from 18 metres.
Luka Modric said Croatia deserved their luck after keeping Russia on the back foot throughout.
"It was a difficult match as we expected but we played a great game. We were a bit lucky with the goal but we earned it because of how we pressed. We were patient and we kept believing until the end," he said.