It was just the third time in NFL history a game was decided on a safety, and the first since 2004.
"I don't know if we could have gone another series, the engine light was on, tyre pressure warnings were going off, I just had to put the pedal to the metal and give it all, let the engine run out," Wake said.
The win also ended a four-game winning streak for the Bengals, who made a furious second half comeback but ultimately could not overcome three interceptions from Dalton.
The Dolphins (4-4) took full advantage of a couple of critical plays in the early stages of the second half to set up a 17-3 buffer.
With a 10-3 lead at half-time following a spluttering Bengals start, Cincinnati had looked poised to tie the game but a fortunate holding call and then a 94-yard interception return produced a 14-point swing in Miami's favour.
Dalton had found Marvin Jones in the clear but his 50-yard touchdown was scrubbed off when fellow receiver Jermaine Gresham was called for holding, an unnecessary penalty given Jones had plenty of space to work in.
The Bengals kept possession but just five plays later, Dalton threw the ball into the arms of Dolphin Brent Grimes and he raced almost the length of the field to open up a 17-3 lead.
The visitors rallied, however, with running back Gio Bernard scoring twice, once from three yards and then a 35-yard effort to tie the scores at 17-17 before Mike Nugent put the Bengals ahead with a 54-yard field goal inside the final two minutes.
Rookie kicker Caleb Sturgis, who had missed an easy attempt early in the game, kept the Dolphins alive and sent the game to overtime with a 44-yard response with 11 seconds remaining.
Both teams failed on two drives each in the extra period before Wake produced the game-winning sack.
"That's huge, the defense played great all day. They put together some long drives but our guys kept fighting and got a few turnovers and when the game counted... Cam Wake steps up and makes a huge play," Miami quarterback Ryan Tannehill said.
"It was a huge win for us. No one likes to lose and losing four straight is terrible. We just needed to right the ship, get back on track and now get ready for next week."
Tannehill threw 20-of-28 for 208 yards and rushed for a touchdown while Dalton ended the night at 32-of-53 for 338 yards and three interceptions.
Lamar Miller rushed for 105 yards for Miami while Bernard had 79 yards and two touchdowns, and A.J. Green claimed 128 receiving yards for the Bengals.
The loss dropped the Bengals to 6-3 but they remain top of the AFC North while the Dolphins sit tied for second in the AFC East.
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