Williams has not won the US Open since 2001 but remained on course for the final with a brilliant display against a resolute Suarez Navarro.
The Spaniard proved a tough opponent for Williams but it was the latter who took the first set 6-3 after getting off to a flying start with a break in the opening game.
However, Suarez Navarro grew into the match in the second set and held her nerve to force a third and deciding set.
And the crowd inside Arthur Ashe stadium were treated to some brilliant tennis as both players went for the win.
Williams was first to break early on with the fans cheering on every point she gained.
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But the unseeded Suarez Navarro was not going down without a fight and was consistently testing Williams on her dominant serve.
A brilliant rally between the two players got the Arthur Ashe crowd on their feet with Suarez Navarro taking the point
But Williams continued her dominance in the third set to once again break serve to put her in a commanding position.
And she held her serve to reach her 12th quarter-final at the US Open of her career.