Afghanistan bowled out Kenya for a mere 93 runs in 43.3 overs with spinners Nabi, Hamza Hotak, and Karim Sadiq sharing seven wickets.
Nabi then scored an unbeaten 46 off 42 balls and hit the winning boundary as Afghanistan raced to 96-3 in only 20.5 overs.
"This is the best day of my coaching life," said a jubilant Kabir Khan, a former Pakistan test cricketer.
Afghanistan Finance Minister Mohammad Omar Zakhilwal, also a member of the national cricket board, said the achievement brought some much-needed happiness to the country.
"Our people have seen years and years of sadness and now we are among the best 12 cricketing nations in the world," he said. "We will encourage businessmen and the government to support cricket."
Afghanistan has played in the last two World Twenty20 tournaments, but to qualify for a World Cup is remarkable progress for a country which has only a handful of turf pitches.
Come 2015, Afghanistan will be in a World Cup group with Australia, New Zealand, England, Sri Lanka, Bangladesh and another qualifier yet to be determined. Two more Associate teams, like Afghanistan and Ireland, will have a chance to qualify in New Zealand next year.
Nabi was quick to warn the big teams in his pool to watch out for Afghanistan.
"Maybe we know about the weaknesses of some teams ... we will do better and do our best to beat some full member teams," he said.
Only five years ago, Afghanistan was playing in World Cricket League Division 5 the lowest ranked tournament among the affiliate members.
But since then Afghanistan has made rapid progress, and its status was raised to an Associate member of the International Cricket Council.
"Afghanistan's journey has been a remarkable one," ICC chief executive David Richardson said in congratulating the team.
UAE also played an important part in the Afghans' progress, as Sharjah has been their home base for at least two years. They have competed against Australia and Pakistan in one-day internationals at Sharjah.
In 2010, Afghanistan qualified for the World Twenty20 in West Indies, and last year played in the same tournament in Sri Lanka. It didn't win a match in either.
Khan said he hoped more prominent cricket nations such as India, Pakistan and Sri Lanka will help Afghanistan prepare for the World Cup by playing ODIs with them.
"It's always helpful when you play against full member countries, unfortunately we don't get much chance to play against them," he said. "I think at least neighboring countries like India, Pakistan and Sri Lanka should help us."
With time to prepare, Khan hopes Afghanistan makes a statement in 2015.
"We just want to leave our mark (at the World Cup) and we want other teams to notice us," he said.
Fast bowler Hamid Hasan helped restrict Kenya to only seven runs in the first 10 overs with some superb swing bowling, and clean-bowled tailenders Nelson Odhiambo and Hiren Varaiya off successive deliveries.
"I've already lost all my words, but I am so, so happy. This is the most happiest day of my life," Hasan said. "To me it's an unbelievable dream come true."
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Amir Shah in Kabul, Afghanistan, contributed.