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.
Three people were injured when a broken-down bus was hit by a lorry on the A34 in Oxfordshire.