Shiffrin's 150th World Cup podium moved her five shy of the record held by Ingemar Stenmark, while her 82nd World Cup slalom podium broke her tie with the Swede for most in a discipline.
Shiffrin won with a combined time of one minute 48.21 seconds to beat 19-year-old Croatian Zrinka Ljutic by 0.14. Anna Swenn Larsson took third.
"I could hear you cheering for Zrinka, which was actually quite cool from the start to hear this noise," Shiffrin told the Slovakian crowd in a finish-line interview.
"I knew she put down an amazing run and I had to push."
Shiffrin's latest victory - the American having broken Stenmark's all-time record when she reached 87 - came a day after Vlhova sustained a season-ending injury.
"We're missing somebody really big today, we missed Petra a lot, so it's wonderful that you stayed," Shiffrin said.
"It was not easy on the second (run), now I feel the energy has gone."
Vlhova tore knee ligaments in a crash during Saturday's giant slalom. The Slovakian and Shiffrin have dominated women's slalom for years, combining for 14 wins in the past 15 races.
"I have been thinking about (Vlhova) a lot the last 24 hours," Shiffrin said earlier on Sunday.
"Honestly, this is just an enormous blow for the sport. It sucks.
"For me personally, over these years, I've grown to love the battles with her, and I think today she would be just so strong, so I really miss watching her ski today and having that battle."
Shiffrin is on pace to claim her eighth World Cup season title in slalom.
