Hogenkamp, the world number 141, survived a match point to win 7-6 (7/4), 5-7, 10-8 in Moscow against world number 17 Kuznetsova, a former two-time Grand Slam champion.
At four hours, the match was the longest singles rubber in Fed Cup history, beating the previous best set in 2005, when Vilmarie Castellvi of Puerto Rico beat Aleksandra Wozniak of Canada in three hours and 49 minutes.
Kiki Bertens, ranked 106, earlier put her country 1-0 up beating 31st-ranked Ekaterina Makarova 6-3, 6-4.
"I'm very happy that I'm the one who could make it 2-0 today," said Hogenkamp.
"And of course it feels great to win the longest match in Fed Cup history.
"I was trying to play my own game and it worked well. It's a good score after the first day but in Fed Cup you still need to win three rubbers so we're not there yet."
Four-time champions Russia now face a dilemma over whether or not to name Maria Sharapova in Sunday's reverse singles despite the world number six insisting a forearm injury would restrict her to the doubles clash which will conclude the tie.
Defending champions the Czech Republic were also in a dogfight in Cluj where they ended the day at 1-1 against Romania.
Two-time Wimbledon champion Petra Kvitova, the world number nine, hit 46 unforced errors in her 6-3, 6-4 loss to Monica Niculescu, the 37th-ranked player.
Earlier, Karolina Pliskova, one of the stars of the Czechs' 2015 final win against Russia, stunned world number three Simona Halep 6-7 (4/7), 6-4, 6-2.
It was 23-year-old Pliskova's first win over Halep in four meetings.
Australian Open champion and world number two Angelique Kerber got Germany's first round match against Switzerland back on track in Leipzig with a 6-1, 6-3 win over Timea Bacsinszky.
"I felt the tiredness in the second set, but the spectators helped carry me through," said Kerber, who was playing her first match since stunning Serena Williams in the Australian Open final last weekend.
The Swiss had taken the lead in the tie when 18-year-old Belinda Bencic, the world number 11, swept past Andrea Petkovic, 6-3, 6-4.
In Marseille, France and Italy were locked at 1-1.
Camila Giorgi defeated Kristina Mladenovic 1-6, 6-4, 6-1 before Caroline Garcia levelled the tie by seeing of former French Open runner-up Sara Errani 6-3, 7-5.
Italy, who have been champions four times since 2006, are bidding to reach the semi-finals for the seventh time in eight years.