Spain, which had drawn its previous two qualifiers played at home, is now undefeated in 29 straight games on home soil.
"It wasn't easy. We weren't very crisp with our final passes, but we got the win and took an important step toward the World Cup," Xavi said.
Xavi wore the captain's armband after Spain coach Vicente del Bosque started Victor Valdes over regular skipper Iker Casillas.
Del Bosque also gave Swansea forward Miguel "Michu" Perez his first intentional start even though he had been a late replacement in the squad for the injured David Villa.
After Spain struggled to create scoring chances in the first 45 minutes, Del Bosque took off defender Nacho Monreal for another playmaker in Andres Iniesta and later sent on Negredo for Michu.
Spain continued to probe and the breakthrough came when Pique blindly lobbed the ball backward from near the end line. After Belarus' Egor Filipenko grazed it, the ball fell perfectly for Xavi to drive in a low right-footed strike.
Negredo secured the result with a header from a cross by Sergio Ramos after Pedro Rodriguez had played the defender free with a deft back-heeled flick.
Negredo dedicated his goal to former Formula One test driver Maria de Villota who was found dead in a Seville hotel early on Friday one year after a life-threatening racing accident. Police said she died of natural causes.
"It's a sad day," said Negredo. "I didn't know her personally, but she was an example of how you can overcome anything."
The game went ahead despite an explosion in the power supply at Iberostar Stadium on Thursday night that injured three people two seriously and led to several power outages.