This is a dynamic list, so it’ll be updated each time I see a film which had its world premiere in 2001 that is better than at least one of the films already on the list.
- La Ciénaga (Lucrecia Martel)
- Mulholland Dr. (David Lynch)
- A.I.: Artificial Intelligence (Steven Spielberg)
- Where Lies Your Hidden Smile? (Pedro Costa)
- The Man Who Wasn’t There (Joel & Ethan Coen)
- I’m Going Home (Manoel de Oliveira)
- Trouble Every Day (Claire Denis)
- The Royal Tenenbaums (Wes Anderson)
- What Time Is It There? (Tsai Ming-liang)
- The Piano Teacher (Michael Haneke)
Other 2001 films I’ve seen
- ABC Africa (Abbas Kiarostami)
- Along Came a Spider (Lee Tamahori)
- Amélie (Jean-Pierre Jeunet)
- America’s Sweethearts (Joe Roth)
- American Pie 2 (J.B. Rogers)
- Atanarjuat: The Fast Runner (Zacharias Kunuk)
- A Beautiful Mind (Ron Howard)
- Black Hawk Down (Ridley Scott)
- Brief Crossing (Catherine Breillat)
- The Devil’s Backbone (Guillermo del Toro)
- Domestic Disturbance (Harold Becker)
- Donnie Darko (Richard Kelly)
- Down to Earth (Chris & Paul Weitz)
- Dr. Dolittle 2 (Steve Carr)
- En construcción (José Luis Guérin)
- Fat Girl (Catherine Breillat)
- Frailty (Bill Paxton)
- Ghost World (Terry Zwigoff)
- Hannibal (Ridley Scott)
- Hardball (Brian Robbins)
- Harry Potter and the Sorcerer’s Stone (Chris Columbus)
- Heartbreakers (David Mirkin)
- Hearts in Atlantis (Scott Hicks)
- Hedwig and the Angry Inch (John Cameron Mitchell)
- Honey for Oshun (Humberto Solás)
- Human Nature (Michel Gondry)
- In Praise of Love (Jean-Luc Godard)
- In the Bathtub of the World (Caveh Zahedi)
- In the Bedroom (Todd Field)
- Jeepers Creepers (Victor Salva)
- Jurassic Park III (Joe Johnston)
- K-PAX (Iain Softley)
- L.I.E. (Michael Cuesta)
- La Libertad (Lisandro Alonso)
- Late Marriage (Dover Koshashvili)
- Legally Blonde (Robert Luketic)
- Life as a House (Irwin Winkler)
- The Lord of the Rings: The Fellowship of the Ring (Peter Jackson)
- The Majestic (Frank Darabont)
- Monster’s Ball (Marc Forster)
- Monsters, Inc. (Pete Docter & David Silverman)
- Moulin Rouge (Baz Luhrmann)
- The Mummy Returns (Stephen Sommers)
- O (Tim Blake Nelson)
- Ocean’s Eleven (Steven Soderbergh)
- Oporto of My Childhood (Manoel de Oliveira)
- The Others (Alejandro Amenábar)
- Pearl Harbor (Michael Bay)
- The Pornographer (Bertrand Bonello)
- Rat Race (Jerry Zucker)
- Rush Hour 2 (Brett Ratner)
- Save the Last Dance (Thomas Carter)
- Scary Movie 2 (Keenen Ivory Wayans)
- Serendipity (Peter Chelsom)
- Session 9 (Brad Anderson)
- Shallow Hal (Bobby Farrelly & Peter Farrelly)
- Shrek (Andrew Adamson & Vicky Jenson)
- Spirited Away (Hayao Miyazaki)
- Storytelling (Todd Solondz)
- Summer Catch (Michael Tollin)
- Sweet November (Pat O’Connor)
- Swordfish (Dominic Sena)
- Training Day (Antoine Fuqua)
- Vanilla Sky (Cameron Crowe)
- The Wedding Planner (Adam Shankman)
- Zoolander (Ben Stiller)

Eclectic list. Mine would include (in order): The Man Who Wasn’t There, Mulholland Drive, Memento, The Royal Tenenbaums, The Claim, Made, Chopper, Eureka, Brother, Ali.
no Martel? I still have to see Man Who Wasn’t There and Chopper.