Suazo scored twice to give Zaragoza a 2-0 win at Getafe in La Liga and both times lifted his shirt to reveal a vest reading 'Fuerza Chile' (Chile be strong).
"It was an important thing that I had to show today," the shaven-headed 28-year-old told reporters.
"I was aware that I needed a goal and that's why I made the vest to give a boost to the whole country," he added.
"A lot of people died in Chile and I am very sad. All my family is there. We cannot talk much. The lines are down. We can only talk in messages."
After Real's 5-1 win in Tenerife later on Saturday, Pellegrini offered his support to the victims.
"I hope the country comes together and manages to overcome this," he told a news conference.
A huge magnitude-8.8 earthquake shook Chile early on Saturday, killing at least 214 people, knocking down homes and hospitals, and triggering a tsunami in the Pacific.
Suazo's gesture recalled one by Cristiano Ronaldo last weekend when the Real Madrid forward revealed a vest with 'Madeira' written on it after scoring against Villarreal.
The Portuguese international's native island had been struck by floods which left at least 42 dead.
Suazo missed a chance to complete his hat-trick late in the match at Getafe and had to leave the pitch after appearing to damage his left hamstring.
