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.
The hosts controlled possession, as expected, but were never at their fluent best and eventually paid the price in dramatic circumstances.
They did look set for victory, however, when Alexis Sanchez scored his 18th league goal of the season with 23 minutes remaining.
But Lafita took advantage of some terrible marking to nod in a late equaliser and stun the home crowd.
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.
His cut-back was met at full pace by the Argentina star, who side-footed a clever finish into the top-left corner from 12 yards.
Barca's lead lasted just 14 minutes, though, as Lafita's outstanding finish restored parity.
An unorthodox lobbed free-kick unsettled the Barcelona defence and Lafita brought the ball down, held off the attention of Pedro and then lashed a shot across goalkeeper Jose Pinto and into the left corner.
The goal was initially ruled out by the linesman, who appeared to suggest that Lafita had handled the ball, but referee Jose Teixeira intervened and the decision was overruled, as the visitors went to the break on level terms after Alexis wasted a late chance.
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 chances kept coming for Barca, as Pedro was denied by Getafe goalkeeper Julio Cesar after an excellent dribbling run from Messi, and their persistence was eventually rewarded in the 67th minute.
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.
With Barca left vulnerable after a quick Getafe counter-attack, Sarabia could only fire a volley straight at Pinto.
But it mattered little, as Lafita's late effort boosted their survival hopes and shattered Barcelona dreams.
