Third seed Stephens, building up to the start of the defence of her maiden Grand Slam title in New York later this month, squandered three match points at 5-1 in the second set.
That setback allowed Suarez Navarro to battle back to 5-5 before the American steadied herself to claim a third win in three meetings against the Spaniard after 90 minutes on court.
Stephens hit 34 winners to the qualifier's 13 and goes on to face either German 10th seed Julia Goerges or Anastasija Sevastova of Latvia.
Australian 15th seed Ashleigh Barty also made the quarter-finals by seeing off Alize Cornet of France 7-6 (7/3), 6-4.
Cornet had knocked out Wimbledon champion Angelique Kerber in the second round on Wednesday.
Britain's Johanna Konta saw off two-time Grand Slam champion and former world number one Victoria Azarenka, 6-3, 6-1, to move into the third round in a tie held over from Wednesday due to rain.
Konta will return to the court later Thursday to face fifth seed Elina Svitolina of Ukraine for a place in the last-eight.
World number one and French Open champion Simona Halep was 3-4 down overnight to Anastasia Pavlyuchenkova of Russia in her second-round tie. That will also be completed Thursday.