Xavi fired the world and European champions in front with a shot that ballooned off a defender's leg and over the keeper after 19 minutes as they dominated possession.
Marius Stankevicius drilled a long-range volley past Iker Casillas for a spectacular equaliser in the 57th minute but any chance of an upset disappeared when Tabas Kijanskas scored an own-goal in the 70th minute.
"The state of the pitch was terrible but the team did their job well and dominated and tried to play their football," Xavi told Spanish television.
"It's all sand. At this level in 2011 to have a pitch in this state, the truth is it's sad for football. These are issues for UEFA and the like but they should come here and see it because you can't play football here.
"We got the points which was the objective. We have one foot and a half in the finals and so we're happy."
Spain made a number of changes from the side that came back to beat the Czechs 2-1 on Friday, but kept faith with David Villa who had scored both goals in Granada to become the country's all-time leading scorer with 46.
Spain played themselves into trouble on the edge of their own area to give Stankevicius the chance to level from a poor clearance, but Xavi helped steady Spain's nerves.
He picked out substitute Mata's run into the area, and the Valencia winger's cross was turned into the net by Kijanskas as Fernando Llorente raced in to connect.
"Luck is very important in football, but one should admit that Spain was stronger today," Spain coach Vicente del Bosque told reporters.
Lithuania boss Raimondas Zutautas praised his side's fighting spirit but also had strong words for the country's authorities over the state of the pitch.
"I hope our government saw how we were playing today," he said. "They would realise that Lithuania needs a national stadium with a pitch which could be warmed up in early spring so we could play despite snow and cold, instead of being ridiculed in front of the whole of Europe on all the television channels."
