Lafita – who initially cancelled out Lionel Messi's first-half strike – headed in a dramatic leveller in the 92nd minute to give Cosmin Contra's side a well-earned point.
Barcelona are now three points behind leaders Atletico Madrid, who have a game in hand on them, with just two fixtures to play.
At the other end of the table, the draw could prove crucial in Getafe's fight against relegation, with the draw moving them up to 16th and three points clear of safety.
Barcelona paid tribute to former coach Tito Vilanova, with a minute's silence held before the first match at the Camp Nou since the 45-year-old's passing.
The hosts also wore commemorative shirts that had words translating to 'Tito – Forever Eternal' emblazoned on them.
Both sides were slow to get going until Messi's 23rd-minute opener, which came after Dani Alves charged down the right.
Barca's lead lasted just 14 minutes, though, as Lafita's outstanding finish restored parity.
Alexis was denied by the crossbar early in the second half, despite failing to meet Alves' swinging cross from the right with his head, almost scoring with his shoulder after significant pressure from defender Rafa.
The goal followed a neat team move, with substitute Cesc Fabregas' ball finding Pedro, and although his shot was blocked, Alexis was on hand to fire the rebound in with his left foot.
Fabregas and Alves had further opportunities to extend Barca's lead, before Messi lobbed onto the roof of the net, while at the other end, Pablo Sarabia was guilty of missing a great chance with 10 minutes left.
But it mattered little, as Lafita's late effort boosted their survival hopes and shattered Barcelona dreams.
