Angel Sepulveda scored a first-half header and Edson Alvarez sealed the win late on at the Alamodome in San Antonio, Texas, as the defending champion scraped to their win.
Goalkeeper Jose Corona starred with numerous strong saves as Curacao – who finish the tournament without a point – threatened often.
With coach Juan Carlos Osorio suspended, an understrength Mexico is yet to impress in the United States, and it will look for improvement against Honduras in its quarter-final.
Mexico was once again much-changed, with Orbelin Pineda the only player to retain his starting spot from the team that played out a dour 0-0 draw against Jamaica.
Coming off a 2-0 loss to El Salvador, Curacao made five changes as Shanon Carmelia, Ayrton Statie, Michael Maria, Quenten Martinus and Gino van Kessel started in place of Quentin Jakoba, Gillian Justiana, Jurien Gaari, Rangelo Janga and Elson Hooi.
Curacao made an excellent start, pushing Mexico – who seemed flat – back in the early stages.
The minnow almost turned its dominance into a 14th-minute lead, but Corona saved well with a strong right hand to deny Van Kessel.
But, against the run of play, Mexico opened the scoring midway through the first half.
Raul Lopez burst clear down the right and crossed to the back post, where Sepulveda headed into the bottom corner.
The goal seemed to settle Mexico, who went close through Martin Barragan and Erick Gutierrez to finish the half.
However, it was again Curacao who started the second half brightly, Leandro Bacuna smashing the crossbar with a header.
Corona made a superb save with his legs to deny Van Kessel again in the 56th minute before Bacuna struck the crossbar from distance.
After the hour-mark, substitute Elias Hernandez almost doubled Mexico's lead, cutting in from the right before seeing his shot tipped over by Curacao keeper Eloy Room.
Mexico should have sealed its win in the 76th minute, but Gutierrez – untracked from midfield – headed a cross wide from near the penalty spot.
Hernandez squandered its next opportunity, stepping past his defender before firing straight at Room from inside the area, and Jair Pereira also hit the crossbar.
But Mexico finally made sure of the result in the 91st minute, substitute Alvarez tucking a finish into the bottom corner for his first international goal.
Meanwhile, Jamaica and El Salvador both booked their place in the knockout rounds after playing out a 1-1 draw in San Antonio, Texas on Monday (AEST).
El Salvador took an early lead through Ruben Marroquin, which was cancelled out in the second half when Darren Mattocks converted a penalty for a Narcisco Orellana's handball in the box.