The European champions had already secured their place in South Africa next year as group winners and coach Vicente del Bosque rested a number of key players including first-choice goalkeeper and captain Iker Casillas.
Xavi led the side in rainy Yerevan and the Barcelona midfielder's chipped pass in the 33rd minute allowed Fabregas to lose his marker and lift the ball over Armenia keeper Roman Berezovski into the net.
Armenia were level on 58 minutes when Carles Puyol appeared to head into his own net, although the goal was credited to defender Robert Arzumanyan, but Spain restored their lead six minutes later from the spot.
Berezovski felled Mata and the Valencia forward converted the penalty low to the keeper's left to seal Spain's ninth win out of nine in the group.
"It was a difficult match and they did not allow us a lot of space," Del Bosque said in an interview on Spanish radio.
"We were missing the final pass on many occasions but we managed to create some good moments and secure an important win," he added.
With David Villa injured, Del Bosque played Fernando Torres as a lone striker and the Liverpool forward made way for Sevilla's Alvaro Negredo to make his first appearance for Spain 10 minutes after the break.
"I am really happy with the debut," Negredo, who was brought into the squad to replace Villa on Tuesday, told Spanish radio. "It was the happiest day of my life and I will never forget it."
Spain's final match in the group is against Bosnia in Zenica on Wednesday.
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