In response to Mr Corbyn's statement, Virgin said: " We're a bit puzzled why Jeremy couldn't find unreserved seats when he boarded the train - they're right next to him as the photo shows."
The company confirmed that two passengers were moved to first class so that Mr Corbyn could sit next to his aide.
Keren Harrison, 45, was travelling on the train and struck up a conversation with the Labour leader when she saw him sitting on the floor.
Mrs Harrison, who is not a supporter of the party, told the Press Association: "I spoke to him and then the staff came over and said 'we'll try and find you some seats, sir' - I don't know if they would have done it for any other passenger - and off they went before coming back and offering him a seat in first class.
"He declined them and I thought he was obviously a man of morals and doesn't want to sit in first class.
"They then came up and said they had managed to find him some seats and I got the impression they had moved some people around."
During the video in which Mr Corbyn is sitting on the floor, he said: "This is a problem that many passengers face every day on the trains - commuters and long-distance travellers.
"Today this train is completely ram-packed. The staff on the train are absolutely brilliant, working really hard to help everybody.
"The reality is there's not enough trains - we need more of them - and they're also incredibly expensive.
"Isn't that a good case for public ownership?"
TSSA union general secretary Manuel Cortes accused Virgin of being "disingenuous" and said Mr Corbyn had " highlighted this daily travel blight which means people sit on floors on trains every day because of overcrowding".
Mr Smith appeared to make reference to the issue in a Twitter post which read: "My campaign remains on track. Proud to be genuinely standing up for ordinary people."