José Mari was shown a straight red card on 22 minutes and Tenerife made the most of their man advantage when Gómez scored just before half-time.
The Real Madrid loanee was on target, once again, just two minutes into the second-half, while Shaq Moore clipped the bar just after the hour-mark.
Tenerife have put together three consecutive victories to move up to eighth. Cádiz have now won just one of their last six matches since the restart and could be overtaken this weekend at the top of the table.
Visitors looking for three important points in Cádiz
Cádiz went into the game a point clear of Real Zaragoza having drawn four of their five games since the restart and started the game brightly, putting early pressure on Tenerife.
The hosts were awarded a free-kick in a dangerous area inside three minutes, but Yann Bodiger’s effort was blocked by the wall and the visitors then started to grow into the game.
With their first attempt in the game, Rubén Baraja’s side went close to taking the lead. Luis Milla delivered a corner into the box, from the right, and Álex Muñoz’s powerful header was cleared off the line by Mari, which denied a certain goal.
Tenerife’s goalkeeper Adrián Ortolá was called into action for the first time on 17 minutes when his low save turned José Manuel Jurado’s curling effort around the post.
Cádiz were reduced to 10-men midway through the first-half when Mari was shown a straight red card for a late challenge on Aitor Sanz. It was the visitors that went on to take control, with the home side willing to sit back and soak up the pressure.
Right on the stroke of half-time, Tenerife made the most of being on top and broke the deadlock. The ball was worked to Moore, whose cross from the right took a slight deflection and was turned home by the sliding Gómez at the back post.
Gómez strikes again at the start of the second-half
The visitors picked up where they left off and doubled their advantage in the game just two minutes into the second-half. Luis Pérez played a dangerous cross into the box from the right and Gómez guided the ball into the far corner of the net for his second of the game.
With the league-leaders trailing and down to 10-men, chances were few and far between since Mari’s sending off. Sergio González went close for Cádiz on 56 minutes, but Tenerife went on to control and look more likely to stretch their advantage further.
Moore beat the offside trap and broke into the box. The American played a low cross straight across the six-yard area before Álex Bermejo struck wide of the post.
Three minutes later, Moore was involved in the heart of the action once more when his cross, which took a slight deflection, clipped the crossbar. Substitute Joselu headed the rebound back across goal, but Cádiz scrambled the ball clear.
With the man advantage, Tenerife controlled the game and didn’t look threatened at all in the final half-hour. Moore had a long-range effort kept out in additional time, but the visitors had already done what they needed to, to pick up an important three points.
Tenerife will head back home for the arrival of Deportivo la Coruña on Tuesday evening, while Cádiz, who have now won just one of their last six matches, play promotion-chasing Elche CF on the road.