Tanak had led overnight after winning the opening two stages but lost power on Friday's opening stage, known as El Chocolate.
"Turbo," said Tanak from the road. "There is a turbo in the car, but it's just dead."
, a Finn who drives for Hyundai, grabbed the race lead as Tanak limped in 7mins 47sec behind.
Frenchman Sebastian Ogier, in a Toyota, who has won the rally six times, took the next stage but the Finn hit back to take the morning's final stage by 0.6sec and stretch his lead to 1.4sec.
"Seb has won this rally so many times and we can match his pace, so I need to be very pleased with that," said Lappi.
Welshman Elfyn Evans was third in a Toyota at 12.5sec.
Reigning champion Kalle Rovanpera, who trails Tanak by three points in the world rally championship standings after two events, was sixth at 29.2sec after suffering mechanical problems following a big slide at the end of the opening stage.
Tanak ended the morning 26th at 12min 35.5sec but was still the best-placed Ford after a nightmare morning for the team's three drivers.
Ford's Frenchman Pierre-Louis Loubet hit a rock on stage three.
Racing was briefly stopped on the stage after private driver Jourdan Serderidis, 59, in the third Ford, stopped, facing the wrong way, in the middle of the road.
Tanak can still pick up bonus points on Sunday's Power Stage.
Japanese driver Takamoto Katsuta went off the road in his Toyota on stage five.
