Obama said in a late-night on-camera statement that the attacks had failed because Americans refused to be terrorized and heaped praise on police and law enforcement services while remembering the dead and injured.
"Tonight, there are still many unanswered questions. Among them, why did young men who grew up and studied here as part of our communities and our country resort to such violence?" Obama asked.
"How did they plan and carry out these attacks? And did they receive any help? The families of those killed so senselessly deserve answers."
Obama also said America owed a great debt to law enforcement services, including the Boston and Massachusetts police forces and the FBI, which tracked down the perpetrators of Monday's attacks.
"These men and women get up every day, they put on that uniform, they risk their lives to keep us safe, and as this week showed, they don't always know what to expect," the added.
The president came to the podium in the White House press briefing room shortly after law enforcement services captured suspect Dzhokhar Tsarnaev, 19, alive in the Watertown suburb of Boston.
He paid special tribute to Massachusetts Institute of Technology police officer Sean Collier, who was killed overnight Friday as the captured suspect and his brother, who later died in an exchange of fire, tried to evade capture.
Obama, who watched the end of the huge manhunt in Boston on television in the White House residence, also remembered the three people killed when twin blasts took place near the finish line of the famed Boston marathon on Monday.
"One thing we do know is that whatever hateful agenda drove these men to such heinous acts will not, cannot prevail," he said.
"Whatever they thought they could ultimately achieve, they've already failed. They failed because the people of Boston refused to be intimidated. They failed because as Americans, we refuse to be terrorized."