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.
Alvaro Negredo opened the scoring with a fine individual effort before the break and Luis Fabiano secured the points at the end.
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.
They were quickly on the back foot against local rivals Getafe at the Bernabeu when defender Raul Albiol was harshly sent off after 28 minutes.
The Spain defender pushed Roberto Soldado, as they tussled for a long ball that was floated towards the home area, but it did not appear to be a clear scoring chance.
The impatient crowd, who had been whistling their own players, swung behind their team and Real improved with 10 men.
Marcelo crossed from the left in the 53rd minute and Higuain controlled on his chest before firing into the top corner.
Three minutes later Getafe were caught in possession in midfield and on a swift counter-attack Higuain doubled the lead.
STANDING OVATION
The 21-year-old Higuain, whose chances have been limited with all the recent arrivals at Real, was denied his hat-trick by the woodwork before being given a standing ovation when he was substituted near the end.
"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.
With the hosts tiring, Barca finally worked an opening when Lionel Messi fed Carles Puyol on the right of the area and the defender crossed low for Seydou Keita to slot in from close range after 72 minutes.
The tireless Mali midfielder, who bagged a hat-trick against Real Zaragoza last weekend, has now scored six goals in La Liga this season.
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.
