The first 15,000 tickets for the 1 January event were made available online at 08:00 GMT.
Organisers introduced an online queuing system and apologised for the problems.
All tickets were claimed within two hours despite the problems. A final 10,000 tickets will be released on 5 November.
Users reported waiting in the queue for around 45 minutes to book tickets.
It is not the first time City of Culture events have caused issues.
High demand for tickets to an acrobatic performance in July also caused the online booking system to crash.
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The City of Culture team said the initial issues on the site were caused by "the volume of web traffic".
Martin Green, director and chief executive officer of Hull 2017, said: "We are aware some users experienced technical difficulties in getting on the site in the first 20 minutes, after which tickets were processed for customers, and normal service resumed."
He also advised people wishing to apply for the second wave of tickets to pre-register their details before 11:00 GMT on Saturday.
The display is due to take place over the city centre from 20:17 GMT, accompanied by a specially commissioned soundtrack featuring music recorded by bands and musicians associated with Hull.
Organisers said the show would be bigger than the New Year's Eve display held in London.
The site offering free tickets for the firework display to mark the beginning of Hull's year as UK City of Culture crashed due to demand.