England struggled to 229-6 on a low and slow Colombo pitch that seemed to be challenging for batsmen.
But Dilshan (108) and Tharanga (102) made a mockery of that, reaching the victory target with more than 10 overs to spare.
The Sri Lankans, who revolutionised limited-overs cricket with their run to the 1996 title in the last World Cup on the subcontinent, will host New Zealand on Tuesday.
India hosts archrival Pakistan at Mohali on Wednesday. The semifinal winners meet at Mumbai on April 2.
England's rollercoaster campaign ran out of steam in the heat and humidity of the Sri Lankan capital.
"We've been thoroughly beaten by a better side today," England skipper Andrew Strauss said. "But we have to be honest, we haven't been good enough during the tournament. This was a step too far."
Sri Lanka captain Kumar Sangakkara praised his opening batsmen, the first pair to combine for more than 700 runs in one World Cup.
"You need a special performance to win big games. Today I think Dilshan and Upul Tharanga combined to give us that little spark of inspiration," he said. "They were absolutely magnificent and ruthless."
But there were some concerns over 38-year-old Muttiah Muralitharan, who hurt his hamstring in the last group match and hobbled during his last two overs for a return of 2-54 on Saturday.
"I think Murali is going to be OK," Sangakkara said of the leading wicket-taker in international cricket. "It happens. He's had a tough couple of weeks so hopefully we'll have him 100 percent for our next game."
Daniel Vettori was heading to Colombo with a lighter bank balance after being fined 90 percent of his match fee for his part in an altercation with South Africa batsman Faf Du Plessis during New Zealand's 49-run upset win at Dhaka on Friday night.
The New Zealand captain and paceman Kyle Mills were found guilty of undue contact with a rival player and conduct bringing the game into disrepute.
Mills was fined his match fee plus 20 percent for his part, despite pleading not guilty and not even being in the playing XI. The injured fast bowler was released later Saturday from the squad to allow for a reinforcement.
The South Africans were headed home to dreaded headlines after continuing their streak of never winning a World Cup knockout match.
"What a choke!" the Citizen newspaper blazed across its front page Saturday, when the Afrikaans-language Beeld bannered its front with the similar message: "Horrible! Super-chokers do it again."
In Mohali, the Indian and Pakistani players are pretty much secluded in their hotel, or at practice.
The India cricket squad arrived in Mohali, where armed soldiers and police lined the roads and traffic was diverted Saturday as Mahendra Singh Dhoni's squad made its way to the same hotel the Pakistan squad checked into on Friday.
An exclusion zone set up around the hotel prevented fans and media from getting close to the players. Security arrangements are equally stringent at the PCA Stadium, where Pakistan held its first training session on Saturday.
"There is no fear in our minds, no feeling that there is any shortcoming when it comes to security," Pakistan allrounder Mohammad Hafeez said. "We are very happy with the system in place, and we are enjoying ourselves."
Police have asked the Pakistan players to give them four hours notice if they plan to go anywhere outside the hotel or practice venue.
Pakistan coach Intikhab Alam wants to keep the focus on sport, not politics.
"Let it remain as cricket and don't make us feel as if we are standing on a warfront," he urged the domestic media. "We don't want to pressurize our players. This is needless."
India Prime Minister Manmohan Singh wants to use the rare match to build bridges, having invited Pakistan counterpart Yousuf Raza Gilani and President Asif Ali Zardari to join him in watching the match.
The 30,000-capacity PCA Stadium has been sold out for day, leaving fans and even politicians clamouring for tickets as price inflate on the black market.
Punjab police have estimated at least 2000 extra police would be deployed in and around the PCA Stadium for the match, Pakistan's first in India since 2007.
India and Pakistan have played at neutral venues since the 2008 attacks on Mumbai when 10 militants from Pakistan killed 166 people.
