Neeson's film earned $13.1 million from Friday through Sunday and finished in second place, according to estimates from tracking firm Rentrak. Comedy "This is Where I Leave You" landed in the No. 3 slot with $11.9 million at domestic theaters.
"Maze Runner" fell slightly short projections for a start of $34 million, the forecast of Box Office Mojo.
The movie stars MTV "Teen Wolf" heartthrob Dylan O'Brien as one of a group of boys living in an isolated paradise where they are trapped by a giant, moving concrete maze. The movie, based on the first novel in a series written by James Dashner, is an attempt to capture the young adult audience that has propelled blockbusters like "The Hunger Games" and "Twilight."
"Maze Runner" opened overseas on Sept. 11, and global sales through Sunday totaled $81.5 million, distributor 20th Century Fox said. The film cost $30 million to make.
Chris Aronson, 20th Century Fox's president of domestic distribution, declared the film "a bona fide home run."
"When you give audiences a film that they want and they like, it's going to do well," said Aronson.
He confirmed that a sequel is in the works. "The Maze Runner: Scorch Trials" is set to open in September of 2015.
In "Walk Among the Tombstones," Neeson plays a private investigator hired by a drug kingpin to find out who murdered his wife. Neeson has earned success as an action star in recent hits "Non-Stop" and "Taken 2." "Tombstones" cost $28 million to make, according to Box Office Mojo.
"This is Where I Leave You," produced for about $20 million, features Tina Fey and Jason Bateman in the story of four grown siblings forced to live together for a week after their father dies.
Last weekend's winner, home invasion thriller "No Good Deed," fell to fourth place, grossing $10.2 million and bringing its total to $40.1 million. Family film "Dolphin Tale 2" took the No. 5 slot with $9 million in its second weekend.
"The Maze Runner" was released by 20th Century Fox, a unit of 21st Century Fox. "Walk Among the Tombstones" was distributed in the United States by Universal Pictures, a unit of Comcast Corp and in Canada by Entertainment One Group. "This is Where I Leave You" and "Dolphin Tale 2" were released by Warner Bros, a unit of Time Warner Inc.
Sony Corp's movie studio released "No Good Deed."