Playing under new coach Cosmin Contra, the home side looked likely to end a winless run stretching back to November when they went ahead first through Ciprian Marica, then Lisandro Lopez and finally Angel Lafita.
But Granada, who had Manuel Iturra sent off, answered every time and goals from Ivan Riki and Youssef El Arabi (two) secured a draw which keeps them three points above their opponents.
The arrival of Contra earlier this week appeared to have the desired effect as Getafe went in front after just six minutes, with Romanian forward Marica finishing into the top right corner from 12 yards.
Granada slowly grew in confidence though, and El Arabi threatened with a close-range header from Fran Rico's corner which he headed past the right post.
Michael Pereira then fired wide for the visitors before a Jeison Murillo header brought a decent save out of Getafe goalkeeper Jordi Codina.
Codina was on hand once more on the half-hour to palm away a powerful drive from Mohammed Fatau, but they eventually got they goal they deserved after 35 minutes with Riki stabbing in a ball from El Arabi.
However, Getafe's response arrived just four minutes later courtesy of a free-kick from Jaime Gavilan, which was headed home from a few yards out by an unmarked Lopez.
Granada once more turned up the heat at the start of the second half, firing a warning to the hosts with a Fatau effort which just missed the target.
It was a warning Getafe failed to heed and they conceded again after 62 minutes when El Arabi made the most of a cross from Brayan Angulo to head home unmarked.
Fortune swung back in the home side's favour just six minutes later though, when Iturra earned himself a second yellow card for a trip on Pablo Sarabia.
And Contra's men pounced on the opportunity within minutes, with Lafita tucking away Sarabia's through-ball.
Yet once again they were unable to stay out in front and woeful defending allowed Granada to pull level after 77 minutes through El Arabi, who nodded in a corner from Rico.
Getafe looked the team most likely to net again thereafter, but Marica was wasteful a minute from time with a close-range header.