The director of the National Security Agency has tried to dampen the current outcry over US government surveillance programmes by insisting such programmes respect Americans' privacy. He also said that he is unable to intercept his own daughters' emails.

General Keith Alexander told a conference of hackers on Wednesday that extensive surveillance had disrupted dozens of terrorist attacks but that technical and policy restrictions protected the privacy of ordinary Americans.

"The assumption is our people are just out there wheeling and dealing. Nothing could be further from the truth. We have tremendous oversight over these programmes. We can audit the actions of our people 100%, and we do that," he said.

Addressing the Black Hat convention in Las Vegas, an annual gathering for the information security industry, he gave a personal example: "I have four daughters. Can I go and intercept their emails? No. The technical limitations are in there." Should anyone in the NSA try to circumvent that, in defiance of policy, they would be held accountable, he said: "There is 100% audibility." Only 35 NSA analysts had the authority to query a database of US phone records, he said.

General Alexander rejected suggestions that his staff could monitor all US internet traffic and phone calls. "The fact is, they don't," he said.

The four-star army general, who in addition to the NSA heads the Central Security Service and US Cyber Command, was responding to mounting concern over alleged civil rights violations by post-9/11 intelligence gathering networks. That pressure mounted on Wednesday when the Guardian published details of a programme called Xkeyscore, which allows analysts to search with no prior authorization through vast databases containing emails, online chats and the browsing histories of millions of individuals.

In Washington, the bipartisan leaders of a powerful Senate committee questioned the truthfulness of the US intelligence community in a heated hearing as officials contradicted previous statements to Congress about the controversial collection of the phone records of millions of Americans.

General Alexander, who is usually shy of publicity, attempted to win over the 7,000-strong gathering of industry professionals in Las Vegas as part of a charm offensive to contain the damage and deter Washington from curbing the programmes.

"Our job is defending this country, saving lives, supporting our troops in combat," he said. "It's our responsibility to provide the information that they need to survive [and] to go after the enemy."

The intelligence community had failed to "connect the dots" to prevent the 11 September 2001 attacks, as well as earlier attacks on New York, US embassies in Africa and the USS Cole, he said. Terrorists driven by a desire to "create a caliphate" of sharia law in the middle east continued to plot attacks. "Terrorists live among us," he said. Expanded and improved intelligence gathering, however, had thwarted 54 terror-related activities, including 13 in the US. Of those 54 activities, 42 were terror plots, he said.

He credited the collection of US phone records' metadata and the interception of the content of foreign communications, saying both programmes helped disrupt a 2009 plot to bomb the New York City subway which would have been the worst attack since 9/11.

General Alexander said NSA operatives worked under strict controls, including oversight by Congress, the executive and judges on the Fisa court, which he said was a tough watchdog: "I can tell you from the wire-brushings that I've received that it is not a rubberstamp."

He also rejected reports of unauthorised, massive snooping. "What comes out is that we're collecting everything. That is not true," he said. The NSA did not sweep up all emails. "That's wrong, we don't. And if we did, we would be held accountable." He did not mention Xkeyscore.

Without naming the whistleblower Edward Snowden, or media outlets which have published Snowden's leaks, the general said the recent revelations would benefit terrorists. "The damage to our country is significant and irreversible." He did not elaborate.

Alexander's foray from the beltway to address hackers at Caesar's Palace had been compared to entering the lion's den. Cyber professionals have a sometimes tense relationship with Washington and have been unnerved by the Snowden revelations and the possible impact on their multi-billion dollar industry. Organisers said they had never sensed this level of tension or apprehension in 16 years of Black Hat. Security guards confiscated eggs – presumably intended to be thrown – minutes before the NSA chief spoke.

A few hecklers interrupted, accusing him of "lying", "bullshitting" and not reading the constitution. "I have read it. So should you," he shot back, earning laughs and applause.

In uniform, General Alexander spoke calmly and sprinkled a few jokes and personal references into his speech, including the fact he has 15 grandchildren. He praised the audience and invited them to help improve NSA. "You are the greatest gathering of technical talent anywhere in the world … I want you to help us make it better." The performance won over the hackers, who applauded warmly at the end.

Alexander, 61, has amassed vast powers since 9/11 by arguing that the US needs to aggressively defend against digital attack. In addition to the NSA, the Central Security Service (which coordinates eavesdropping between the armed services) and US Cyber Command, which was set up under the Obama administration to deter cyber-attacks, he also commands the navy's 10th fleet, the 24th Air Force and the Second Army.

A Wired profile said Alexander inspired within the government a mix of respect and fear not seen since FBI director J Edgar Hoover. Nicknames include Emperor Alexander, for his ability to expand bureaucratic fiefdoms, and Alexander the Geek, for his longstanding interest in electronics and computers. The walls of his headquarters inside Fort Meade in Maryland are reportedly wrapped in protective copper shielding, to block giveaway electromagnetic signals.

The Pentagon is enduring deep cuts but Alexander persuaded it in April to seek $4.7bn – a $1bn increase – in the 2014 budget for cyber-operations. That could be imperilled by eroding public and political support following Snowden's leaks.

A Pew Research Center survey found that 47% of Americans worry that anti-terrorism surveillance programmes go too far and endanger civil liberties, against 35% who worry they don't go far enough. In 2010 only a third thought such programmes went too far.

A congressional alliance of libertarians on the right and civil rights advocates on the left came within a whisker – seven votes – of passing an amendment to curb bulk collection of phone records.