Handed a rare start ahead of fellow strikers Fernando Torres and David Villa, Soldado fluffed a chance in the first half but made no mistake from close range when set up by substitute Cesc Fabregas in the 86th minute.
It was cruel on the home side, who had held out stubbornly against their illustrious visitors despite seeing little of the ball and losing goalkeeper Giorgi Loria to injury with some 20 minutes from time.
They even came close to taking a shock 54th-minute lead when Aleksander Amisulashvili's curling shot hit the post but Spain's relentless attacking eventually wore them down.
"They closed ranks from the start and we tried everything but we were missing a little bit of attacking dynamism," Spain coach Vicente del Bosque told Spanish TV.
"These are three exceptionally important points because if we had drawn today it would have been a significant setback," he added.
Spain's victory in their first Group I game extended the Iberian nation's winning streak in European Championship and World Cup qualifiers to 23 matches. They last failed to win in a 1-1 draw with Iceland in Reykjavik in September 2007.
FAMILIAR PROBLEM
The latest crop of Spanish players were confronted by a familiar problem in Tbilisi as tried to penetrate the massed ranks of opposition defenders while dominating possession.
David Silva crashed a long-range effort off a post in the 28th minute before Xavi also tried his luck from distance and forced a fine save from Loria.
Silva and Andres Iniesta nearly found a way through several times with some slick interplay but Spain had to wait until the 45th minute before they finally created a clear chance.
However, when Xavi threaded the ball through to Soldado the onrushing Loria blocked his low shot.
After the woodwork saved Spain early in the second half, Del Bosque went in search of a winner by throwing on Pedro, Santi Cazorla and Fabregas who conjured a low cross that Soldado turned clinically past substitute keeper Roin Kvaskhvadze.
"It was a very hard-fought win," Soldado told Spanish TV.
"It was very tough to find space and create chances but at the end we managed to get the breakthrough," added the striker who took his tally to four on his sixth appearance for Spain.
