Thorpe Park had offered a 12p entry price to a limited number of people who logged into its site at 10:00 GMT on Good Friday using a promotional code.
Hundreds of people complained on social media that they were met with a sold out message at the stroke of 10:00 GMT.
The Surrey attraction said technical difficulties were to blame.
It confirmed later that all 1,871 discounted tickets were sold by 10:00 GMT.
Emma Elms described the situation as an "absolute joke" on the theme park's Facebook site.
"I uploaded my details at 09:59 to be told they had sold out, absolute joke, if you are going to advertise tickets at a selling time of 10am you should at least stick to that," she said.
Another unhappy customer, Rachel Rose, said the website displayed the sold out sign at 09.57 GMT and she was "absolutely fuming".
"Why advertise a selling time of 10am if all of them are sold out prior to that? So disappointed with Thorpe Park today - good luck to you on all of the phone calls and emails you will be receiving from unhappy customers."
Thorpe Park later tweeted: "Due to tech difficulties our sale went live a few minutes sooner than anticipated.
"Sorry for any inconvenience caused."
The theme park also posted: "We understand everyone's frustration with our technical difficulties today. We are replying to all tweets and thank you for your patience."
Some customers who got one of the 12p tickets also took to Twitter to boast of their success.
Entry to the attraction in Chertsey usually costs from Â£27.99 per person online.
In a statement, Thorpe Park said the ticket discount was organised to celebrate the arrival of its new ride Derren Brown's Ghost.
"A problem with an increase in the bandwidth to the website meant the sale went live "two-three minutes earlier than expected.
"All tickets were sold and email confirmations sent to the winners of the 1,871 tickets by 10am.
"Sorry to those that were unsuccessful, and congratulations to those who secured a 12p ticket."
A theme park has had to apologise to disappointed customers after heavily discounted bank holiday tickets went on sale sooner than advertised.