The sanctions come after North Korea's rocket launch last week.
Republicans had been blocking the State Department nominees on behalf of Cruz.
The House will take up sanctions legislation against North Korea.
The Senate will take a procedural vote on Feb. 22.
Rep. Labrador argued House GOP leaders are bad negotiators.
Cristina Marcos 02/11/16 05:11 PM EST
The vote came after the GOP decided not hold hearings on Obama's final budget.
Sen. Mike Lee objected on behalf of Rubio, who was in South Carolina.
The legislation was stalled over a fight on an Internet tax ban.
Defense hawks are calling for extra funding.
The Senate is hoping to pass the customs bill Thursday.