The board found that while Paterno fulfilled a legal obligation to tell his superiors that an employee claimed Sandusky abused a young boy in a shower, it said Paterno should have done more.
"We determined that his decision to do his minimum legal duty and not to do more to follow up constituted a failure of leadership by Coach Paterno," the trustees wrote.
The report comes after months of criticism from Penn State alumni over Paterno's firing in November. The Hall of Fame coach died in January after a brief bout with lung cancer.
Other critics, including some faculty, have questioned the board's decisions in general in the frantic weeks after Sandusky was arrested Nov. 5.
Paterno's lawyer, Wick Sollers, said in a statement later Monday that the family was surprised and saddened that the board "believes it is necessary and appropriate to explain — for the fourth or fifth time — why they fired Joe Paterno so suddenly and unjustifiably on Nov. 9, 2011."
The board conducted a series of media interviews in January about its decisions to remove Paterno and President Graham Spanier.
Trustees interviewed by The Associated Press had cited three reasons for Paterno's immediate removal: a failure to meet a moral obligation in not doing more to report the 2002 allegation; concern that statements from Paterno had challenged the board's authority; and worries that Paterno would not be able to properly represent the school if allowed to finish the 2011 season.
In their statement Monday, the trustees said they had been asked by the Penn State community to "state clearly" the reasons for Paterno's dismissal and the removal of the university president.
"Many people have indicated that they did not understand, and this is our last attempt to try to make it as clear as possible," Trustee Keith Eckel told The Associated Press on Monday. "And people are welcome to agree or disagree with us."
Sollers called the statement another attempt by trustees "to deflect criticism of their leadership by trying to focus the blame on Joe Paterno. This is not fair to Joe's legacy; it is not consistent with the facts; and it does not serve the best interests of the University.
"The Board's latest statement reaffirms that they did not conduct a thorough investigation of their own and engaged in a rush to judgment," Sollers said.
Sandusky is accused of sexually abusing 10 boys over a 15-year span. He has denied the allegations.
Then-graduate assistant Mike McQueary's claim that he saw Sandusky sexually abusing a young boy inside a football building on the university campus is one of 10 such allegations brought by the state attorney general's office.
The first round of charges against Sandusky was filed Nov. 5, four days before Paterno was fired and Spanier was forced to resign.
The board also again apologized for the decision to fire Paterno by phone late that night — a decision that drew the ire of many of the coach's supporters.
"We saw no better alternative," the trustees wrote. "Because Coach Paterno's home was surrounded by media representatives, photographers and others, we did not believe there was a dignified, private and secure way to send Board representatives to meet with him there."
The trustees said they planned to apologize to Paterno for the way he was being dismissed but the coach ended the call before the message could be delivered.
Bitterness over Paterno's removal has turned up in many forms, from online postings to a note placed next to Paterno's statue at the football stadium blaming the trustees for his death.
An unprecedented 86 candidates are on the ballot for three open, alumni-elected seats up for election this spring. Many of the candidates for the 32-member board have said they are running out of unhappiness over its actions in November.
Candidate Mark Connolly said he found it "very, very difficult" to believe that trustees received no information about the case from 2002 on.
"I would submit that if they did receive any information, then they are the ones who displayed a lack of leadership and should have taken more aggressive action at that time, and not wait until indictments in November 2011," Connolly said in an interview.
The trustees said they had intended to name Paterno head coach emeritus in honour of his contributions to the university. The board said additional options are under review but no decisions will be reached until after independent counsel and former FBI director Louis Freeh issues a final report on a special investigation into the school's handling of the allegations.
Freeh's report isn't expected to be released until this fall at the earliest.
The board said Spanier's removal was also a result of a "failure of leadership" that included insufficiently informing the board of his knowledge of the 2002 allegation. Spanier also made public statements that were not authorized or contrary to the board's instructions, the trustees said.
In one statement following the charges against Sandusky, Spanier expressed his "complete confidence" in a pair of university administrators, athletic director Tim Curley and senior vice-president Gary Schultz, who were charged with perjury and failure to report suspected abuse.
Curley is on administrative leave and Schultz retired after being charged. Both maintain their innocence.
Spanier requested a vote of confidence from the board on Nov. 9, according to the statement Monday, and was ousted that night.
But there appears to be conflicting information from Penn State and trustees about Spanier's departure.
The board announced after its Nov. 9 meeting that it decided Spanier was "no longer president." Neither the board nor Spanier had specified then if he resigned.
A school spokeswoman confirmed the following week that Spanier had actually offered his resignation to Penn State earlier in the day. It's unclear whether his resignation was accepted, or whether Spanier was let go regardless.
Trustees in January told the AP his contract as president had been terminated. Spanier remains a tenured faculty member and is on sabbatical.
Also Monday, a judge heard arguments but opted not to rule immediately on how much information Sandusky should get in advance of his trial on child sex abuse charges.
Judge John Cleland gave no indication when he'd rule on the arguments concerning the "bill of particulars" about the 10 purported victims that the attorney general's office provided Sandusky's attorney two weeks ago.
Pennell reported from Philadelphia. Mark Scolforo in Bellefonte, Pa., and Michael Rubinkam in Allentown, Pa., contributed to this report.