The Teessiders insist they requested more tickets for the game at the Stadium of Light on Sunday, January 29, but trouble at previous derby matches prompted Sunderland's safety advisory group to order restrictions.
A statement released by npower Championship side Boro said: "We sympathise with fans who were unable to purchase tickets for the FA Cup tie with Sunderland."
It continued: "Unfortunately, despite strenuous attempts on our behalf, the club's allocation was limited to just 3,000 tickets.
"We were very much aware of the strong demand for this fixture and argued for a significant increase on this figure.
"Sunderland's safety advisory group cited problems at the Stadium of Light in previous derby matches as the reason for the small figure.
"Indeed, initially they were arguing for an even smaller figure of 2,700 tickets, the same figure we were given at our last game where we know significant problems were faced by our fans.
"We managed to obtain a 10% increase on this figure, although we accept the allocation still falls significantly short of demand."
Boro's allocation went on sale on Saturday morning and swiftly sold out.