Spain, who started qualifying with a 1-1 home draw with Greece in midweek, trailed in Georgia at half-time after Khvicha Kvaratskhelia’s 43rd minute strike.
But La Roja found an equaliser 10 minutes into the second half when a sliding Ferran Torres bundled a low cross home.
And in stoppage-time Georgia goalkeeper Giorgi Loria couldn’t hold Daniel Olmo’s high shot which went in off the frame of the woodwork.
Spain play Kosovo in their third qualifier in midweek and manager Luis Enrique admitted he has concerns after their slow start.
“I am very worried about the Kosovo game,” he said.
“If we lack football, precision, finesse or freshness, we will play with all that. Heart, head or whatever, but Spain have to be at the next World Cup and we are not contemplating not being there.”
Elsewhere in UEFA World Cup qualifiers on Sunday, Serge Gnarby scored the only goal of the game as Germany edged Romania 1-0, while Italy won 2-0 in Bulgaria.
England got past Albania 2-0 in Tirana while France responded after their 1-1 opening draw with a routine 2-0 win in Kazakhstan although Kylian Mbappe had a penalty saved.
Denmark thrashed Moldova 8-0, Armenia shocked struggling Iceland 2-0, Sweden cruised 3-0 over Kosovo and Switzerland got past Lithuania 1-0.