Kucherov notched the primary assist on Artemi Panarin's goal at 13:55 of the third period that iced the quarterfinal victory for Russia. Kucherov finished with three shots and was plus-two.
Bolts forward Vladislav Namestnikov blocked a shot with his left foot on the penalty kill in the opening period and had to leave the game. He played just 2:40 although he was spotted on Team Russia's bench later in the game.
Russia moves to Saturday's semifinals where it will play Canada.
Russia is searching for its first goal medal at the World Championship since 2014 and has medaled in each of the last three years, including silver in 2015 and bronze last year.
Canada 2, Germany 1
Jon Cooper-led Team Canada held off host Germany for a 2-1 victory at Cologne's LANXESS Arena in the quarterfinal round of the World Championship.
Canada outshot Germany 50-20 but had to stave off a last-gasp effort by the Germans over the closing minutes.
Mark Scheifele got Canada on the board at 17:11 of the opening period after netting a power-play marker. Jeff Skinner made the score 2-0 Canada with his fourth goal of the tournament, which came with 1:52 remaining in the second.
Germany netted a shorthanded goal with 6:39 remaining in the game when Yannic Seidenberg was able to put a puck past Canada's goaltender Calvin Pickard. The Germans were unable to find the equalizer, however.
Lightning forwards Brayden Point and Alex Killorn were held without a point in Canada's win. Point registered one shot in 12:03 of ice time. Killorn notched five shots in 13:57.
Canada and Russia will battle on Saturday in the tourney's semifinals.
Sweden 3, Switzerland 1
William Nylander's goal at 13:15 of the second period broke a 1-1 deadlock and propelled Team Sweden into the semifinals with a 3-1 victory over Switzerland at Paris' AccorHotels Arena.
Nylander had a goal and an assist to lead Sweden.
Nickas Backstrom netted his second goal in three tournament games to open scoring at 4:15 of the first period. Switzerland leveled the score 1-1 on Gaetan Haas' goal at 12:53.
After Nylander broke the tie, Alexander Edler added an insurance goal at 3:44 of the final period to secure a spot in Saturday's semifinals for the Swedes.
As he's done throughout the tournament, Tampa Bay defenseman Victor Hedman led Team Sweden for time on ice against Switzerland, logging 24:55 and one shot. Fellow Bolts blueliner Anton Stralman played 23:52, second most on Sweden, which will go up in Saturday's semifinal against Scandinavian rival Finland, a surprise winner over the United States on Thursday.
Finland 2, United States 0
The United States completed preliminary play as the top team in Group A but was unable to continue its strong play in the quarterfinals, falling 2-0 to Finland at Cologne's LANXESS Arena.
Harri Sateri put up 26 saves to record his first shutout of the tournament.
After a scoreless opening period, Mikko Rantanen broke the ice with a power-play goal just 1:01 into the second period to put Finland up 1-0.
Joonas Kemppainen added a second goal for the Fins off a two-on-one rush at 6:49 of the third.
Team USA finished the World Championship with a 6-0-0-2 record after tying a U.S. mark with six-consecutive wins in the preliminary stage.
