Spain, already assured of top spot, finished with 21 points after seven wins in eight matches, having lost only to second-placed Scotland, who had also already qualified for the finals and ended with 17 points after a 3-3 draw with Norway.
Georgia finished fourth with eight points, three behind Norway in third, while Cyprus were bottom without a point.
But for Spain coach Luis de la Fuente it was the most bitter victory of his career after Gavi, who plays for Barcelona, left the pitch in tears during the first half while grabbing his knee in pain.
"It is a very difficult moment for Gavi, Barcelona, the national team and for me, it seems as if we have lost the game," De la Fuente told a news conference.
"This is the ugly part of football. I am very sorry. Let's wait for the tests to confirm the extent of the injury. This is the most bitter victory I have ever experienced in my life."
Spain grabbed the lead in the fourth minute after a free kick from Ferran Torres on the left was headed home by Robin Le Normand for his first international goal.
Georgia levelled six minutes later following a quick break and a simple finish from forward Khvicha Kvaratskhelia before Gavi was injured in a collision with an opponent.
"We can't celebrate as we would have liked. Gavi's injury is a hard blow, I'm really gutted," said Le Normand. "It wasn't my best game with the national team, there are things to improve".
The hosts restored their lead 10 minutes into the second half with a header from Torres, who celebrated by holding up the number nine jersey of his Barca teammate Gavi.
Georgia's Luka Lochoshvili put the ball in his own net in the 72nd minute to seal the Spaniards' win.
"An important victory to end on a good note, but we leave with a bittersweet feeling because of Gavi's injury," said Torres.
"We don't know the extent of the injury, but knowing how much of a warrior Gavi is, we think it could be serious."
