Real have 22 points from nine games and are one point behind la Liga leaders Barcelona after the champions were held to a 1-1 draw at Osasuna, Gerard Pique scoring in his own net in the last minute.
Sevilla are third with 19 points after seeing off struggling neighbours Xerez 2-0.
In the late game Athletic Bilbao rode their luck to spoil Quique Sanchez Flores's first league match as Atletico Madrid coach by winning 1-0 at San Mames.
Javi Martinez's first-half header did the damage, though Atletico twice hit the woodwork in the closing stages.
Pellegrini's position was called into question after Real's embarrassing 4-0 defeat by third-tier Alcorcon in the King's Cup in midweek made it three games without a win.
Three minutes later Getafe were caught in possession in midfield and on a swift counter-attack Higuain doubled the lead.
STANDING OVATION
"It has been a difficult week and we had to fall back on our pride," director general Jorge Valdano told reporters.
"The team did well to overcome their early nerves and a tough sending off. With so many apocalyptic commentaries around our message was to respond on the pitch and the players have understood this perfectly."
Barca became bogged down in the face of Osasuna's pressing game at the Reyno de Navarra and they failed to break out of their straitjacket until late in the second half.
Barca appeared to be cruising to victory but in the last move of the game Osasuna's Javier Camunas crossed from the left and Pique gifted the home side a well-deserved point.
