The islanders had made a strong start forcing home keeper Miguel Angel Moya into two sharp saves before Navarro flew into a challenge on Diego Castro.
The referee showed a straight red card, but Getafe, who started the game in the relegation zone, had to wait until the 49th minute to take the lead when Castro hammered a shot high into the net after a pull back in the area.
Joaquin Caparros's visitors never gave up and Javier Arizmendi had the chance to level in added time, but he fired wide with only Moya to beat.
Mallorca slipped down to fifth with 11 points from six games, seven behind leaders Barcelona who maintained their perfect start to the campaign with a thrilling 3-2 comeback win at fourth-placed Sevilla on Saturday.
Champions Real Madrid, who thumped Deportivo Coruna 5-1 at home on Sunday, are sixth with 10 points, three ahead of Getafe in 11th.