Tannehill came out after hurting his left quadriceps on a sack by New York's Calvin Pace, but Moore was up to the task, completing 11-of-19 passes without a turnover.
The Dolphins defense also came up big, sacking Jets quarterback Mark Sanchez four times and forcing a pair of turnovers for Miami (4-3), which racked up its third straight win.
Olivier Vernon scored on a blocked punt and Daniel Thomas added a rushing touchdown for Miami.
Sanchez went 28-of-54 through the air for 283 yards with a TD and an interception as New York (3-5) fell into last place in the AFC East.
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