Brazilian defender Adriano scored twice in an embarrassingly one-sided contest as the Spanish and European champions set up a final with Brazil's Santos on Sunday.
The gloss was taken off Barca's victory in front of a crowd of 66,000 when Spain striker Villa fractured his left leg under a heavy challenge in the first half.
"We're all very sad for David," said Barca coach Pep Guardiola, whose side are bidding to win their second Club World Cup in three years.
"He will be out for a long time with this sort of break. It's a massive blow for him and Barcelona.
"He will return to Barcelona as soon as possible for an operation, escorted by a physician. We will have to win the Club World Cup without him but he is in our hearts."
A glum-looking Guardiola added: "We didn't give Al Sadd a chance today."
Barca took the lead in comical fashion after 25 minutes when Pedro's cross bobbled off Al Sadd defender Nadir Belhadj and goalkeeper Mohamed Saqr's panicked clearance bounced off Adriano's shin into the back of the net.
Adriano scored his second goal two minutes before half-time, latching on to a pass from Thiago Alcantara to drive a powerful, left-foot shot into the bottom corner.
Argentine wizard Lionel Messi then brilliantly picked out Seydou Keita, who calmly flicked home Barca's third goal in the 64th minute with Asian champions Al Sadd chasing shadows.
SECURITY GUARDS
Substitute Maxwell completed the rout after 81 minutes as Barca's quick passing game befuddled Al Sadd, who gave the Catalan giants less trouble than stadium security had on their arrival for the match.
Messi and several Barcelona team-mates looked confused at being held up by zealous Japanese security guards, after getting off the bus, for not having their accreditation visible.
"It's terrible what happened to Villa," said Barca midfielder Javier Mascherano. "We're all sad about what happened and want to offer him our support.
"Today we calmed down after the first goal and the second goal almost finished the game."
Guardiola, whose thoughts were clearly on the stricken Villa, said he was relishing the chance to take on Santos in Sunday's mouth-watering final in Yokohama.
"Santos have a great history - they produced Pele," said Guardiola. "They are still producing great players and will be very dangerous."
Al Sadd coach Jorge Fossati said: "It was a rewarding experience for my players to play against the best team in the world.
"Barcelona have a kingdom and there is no team that can topple that kingdom."
Santos, led by Brazil hot-shot Neymar, will provide an altogether different test, having beaten Japan's Kashiwa Reysol 3-1 in Wednesday's first semi-final in Toyota.
