Bolatti netted five minutes after coming on as a substitute to clinch Argentina the fourth automatic qualification spot from the South American region, joining already-qualified Brazil, Paraguay and Chile.
Uruguay now face a two-legged play-off against the fourth-placed team from North America, Central America and the Caribbean.
The visitors had started the match in danger of missing out on the World Cup for the first time since 1970. A loss to Uruguay coupled with an Ecuador win in Chile would have produced that scenario.
However, Maradona was celebrating wildly on the touchline after Bolatti's goal guaranteed them a spot in the finals next year in South Africa.
The talisman of Argentine football wept and hugged members of his coaching team in an outpouring of emotion at the final whistle.
Having replaced Alfio Basile as coach midway through the qualification campaign, Maradona's credentials for the job had been widely questioned after his side lost to Bolivia, Ecuador, Paraguay and at home to Brazil.
"I want to dedicate this to all the Argentines," he told Argentine television. "I thank the squad."
Ecuador's hopes of qualifying for a third successive World Cup were killed off on Wednesday when they lost their final game 1-0 away to Chile.
Striker Humberto Suazo scored early in the second half for Chile, who had already secured their place at the World Cup finals for the first time since 1998.
Shaven-headed Suazo fired the ball into the roof of the net from about six metres after Rodrigo Millar touched Arturo Vidal's pass into his path.
Ecuador midfielder Segundo Castillo was sent off for a shocking challenge in the 85th minute when he landed his studs in a Chilean player's groin.
Although they needed a win to have any realistic chance of overhauling either of Argentina or Uruguay, Ecuador showed little urgency in the first half.
Edison Mendez had their best chance when he scraped the post after a one-two with Cristian Lara just before the break.
Ecuador, who lost 2-1 at home to Uruguay on Saturday, were more adventurous after Suazo's goal but they also left gaps at the back and were lucky not to go further behind when Jorge Valdivia's chip was turned over by goalkeeper Marcelo Elizaga.
Chile ended their 18-match campaign with 33 points and Ecuador mustered 23 points to finish sixth in the 10-team group.
