Cristiano Ronaldo's own goal early in the first half was enough to hand the former Recreativo and Racing Santander boss a win on his debut in the Andalucians' hotseat and ensure his new troops distanced themselves from the drop zone.
And Alcaraz, who replaced Juan Antonio Anquela as coach earlier in the week, was quick to pay tribute to Granada's resilience against their more fashionable opponents, but called for calm from their supporters in the wake of their triumph.
"I was the one who had said this week that we could win, so I was surprised less than others with the result," the 46-year-old told reporters after the home victory.
"I have not made history, but it was a day of celebration for Granada. Maybe we did not see lots of the ball, but the attitude and solidarity of the players deserves credit.
"These games demand that you give almost all of your physical strength and attention, but we cannot delude our fans and their expectations [shouldn't soar]."