Fifty people were on the First Great Western rail replacement service from Oxford to Didcot when it was hit at Drayton, at 00:30 GMT.
One of the three people taken to hospital sustained serious but non life-threatening injuries, police said.
The southbound carriageway has been reopened but investigations are continuing.
The remaining passengers, who did not require hospital treatment, were taken to Abingdon police station.
Passenger Rebecca Crow posted a picture of her injuries on Twitter and wrote: "I was on the bus, it was insane. Very thankful for no fatalities; just a bust up face!"
The bus was a replacement for the 00:05 First Great Western service from Oxford to Didcot.
Bus passenger Adam Graves said on Twitter: "One moment the bus power failed, a minute or 2 later the lorry hit us. Bus ended up part down the embankment. #a34"
He later told BBC News: "It sort of jumped forward. There were people screaming inside.
"Then you suddenly noticed you're falling out of your seat - everyone was flying everywhere.
"Once that initial shock had passed people were quite calm.
"A lot of people on the bus had been coming back from a gig in Oxford, that's where I'd been, that kept people calm as a few recognised each other."
In a statement, a First Great Western spokesman said the firm was "keen" to offer support to all of the bus passengers and their families.
He said initial reports from its own contractor "suggest the driver followed correct safety procedures", but added police were investigating.
"We are offering them any assistance we can to understand the full details of the incident," he said.
The A34 was shut between the Botley and Milton interchanges.
It had been due to reopen at 11:00 but was delayed due to difficulties in recovering the vehicles, police said.
Officers on site were having problems earlier with drivers ignoring the closure and driving through, according to BBC Radio Oxford travel reporter Sara Dumbell.
She said diversions had also led to lengthy traffic delays in the area.

The former Bond girl has been involved in early workshops for the production from a script by One Man, Two Guvnors writer Richard Bean.
It is understood Arterton would be in line to play the lead role of Rita, played in the film by Sally Hawkins.
Producers Stage Entertainment told the BBC they would "very much love her to be part of the production".
It has not yet been announced which London theatre the musical will be staged at, but it is hoped it will play next year.
"As with all casting there are schedules to work out, but we'd very much love [Arterton] to be part of the production when it opens in the West End," a spokesman said.
Early workshops were directed by Rupert Goold - the new artistic director of London's Almeida Theatre - with the score written by David Arnold, who also provided the music for the Bafta-nominated film.
Made in Dagenham told the story of the machinists who went on strike in 1968 at the Ford Dagenham car factory demanding equal pay and protesting against sexual discrimination.
The character of Rita played a pivotal role in leading the strike and meeting former employment secretary Barbara Castle, leading to the Equal Pay Act 1970.
The film also starred Andrea Riseborough, Jaime Winstone and Bob Hoskins.

The FBI seized computers and mobile phones from the home in October, after celebrities including Jennifer Lawrence had pictures stolen - apparently via a hack on their Apple iCloud accounts.
Court papers have identified eight celebrities by their initials only.
The FBI said no charges have been filed at the current time.
Last summer's scandal highlighted the vulnerabilities of cloud computing - which allows personal information to be stored on the Internet and accessed from any device.
The affidavit - a sworn statement to the court - has identified eight celebrities by their initials only: AS, CH, HS, JM, OW, AK, EB and AH.
One of the identified stars learned she was a victim of the leak through the news media. The papers have stated that "all photos were taken with her iPhone and sent through iMessage to her boyfriend".
Actress Jennifer Lawrence has since spoken about the leak. Speaking to Vanity Fair, the 24-year-old said: "It is not a scandal. It is a sex crime. It is a sexual violation. It's disgusting."
Three people were injured when a broken-down bus was hit by a lorry on the A34 in Oxfordshire.
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Gemma Arterton is set to star in a West End musical version of the 2010 film Made in Dagenham.
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