"Okay, there you go. Beautiful!" radioed astronaut Michael Collins as the two parts of Apollo 11 undocked.
Collins would orbit the Moon in the command module Columbia while crewmates Armstrong and Aldrin descended to the surface in the lunar module Eagle, its flight controlled by an onboard computer.
At the start Eagle was flying feet first, with its face to the Moon.
While Aldrin focused on the computer, Armstrong looked out his small left-hand window for known landmarks, timing when they appeared.
They were coming up early, meaning Eagle was further along its track than planned.
The crew fired Eagle's main engine to bring the craft down to an altitude of 15km.
When it reached that point the computer fired the engine again, braking hard to start the craft curving downward.
In less than 13 minutes Eagle would make a controlled fall to the Moon's surface and come to rest in the Sea of Tranquillity.
The crafthad to land on flat ground or it might not be able to lift off again.
Four minutes into the engine burn Armstrong rolled Eagle over onto its back with its windows looking out to space.
Earth was right ahead but there would be little time to enjoy the view.
Two minutes later the computer flashed up an alarm with code 1202, something the astronauts had never seen in training.
Suddenly they needed to know if the landing could go ahead.
Armstrong paged Houston urgently and checked the button that would abort the flight.
Houston quickly decided the computer was complaining of overload and if the alarm did not occur too often it would be safe to proceed.
The computer had been tied up by signals from one of the craft's radars - a fact that would come to light only later.
Meanwhile Eagle was still plunging to the Moon's surface and more alarms would go off - and each time they did, the computer rebooted.
Around 1500 metres the computer triggered the second phase of the landing, tilting the craft more upright and starting to point its legs towards the Moon, which was now well in view through the windows.
"Eagle, Houston. You're GO for landing," Houston called.
Around 900 metres the computer flashed up three alarms within a minute.
Busy clearing alarms, the astronauts had no time to check the ground.
When they did, Armstrong looked out the window and found navigation errors had taken them six kilometres past the planned landing spot.
The automatic landing program was flying them down towards a crater as big as a football field and surrounded by car-sized boulders.
They were 180 metres up, descending at six metres a second.
Armstrong took the controls and started flying manually, his eyes fixed out the left window.
He slowed Eagle's descent and flew the craft along the north side of the crater while Aldrin called out instrument readings: "400 feet [altitude], down at 9 [feet per second], 58 [feet per second] forward."
The rock field stretched out around the crater and Armstrong could not yet see a place to land.
Tension was growing in Houston; Armstrong had never flown like this in the simulator.
"Looks like a good area here," he said at 80 metres, only to change his mind a little later.
Thirty metres above the ground the low-fuel light flicked on.
"75 feet ... down a half, 6 forward," Aldrin called.
"60 seconds," Houston said.
They had 60 seconds' worth of fuel before they had to either land within 20 seconds or fire the ascent engine and blast back into space.
But below 15 metres there would be no time to fire the ascent engine, so if they ran out of fuel they would crash.
Eagle was now 12 metres above the ground, its exhaust kicking up a fog of dust that made it hard to judge the craft's sideways motion.
"30 feet [altitude], two-and-a-half down," Aldrin called.
Armstrong had picked the final spot for touchdown.
"30 seconds," Houston warned.
They were crawling down now.
"Contact light," Aldrin called. One of Eagle's foot-probes had touched the ground.
Eagle dragged slightly to the left then settled down gently like a helicopter.
With a flurry of actions Armstrong shut off the engine.
It was done. They were down.
The dust blasted sideways by Eagle's engine raced out and disappeared, and in an instant the Moon's surface lay still again.
Armstrong and Aldrin shook hands without a word.
Mission Control had seen the engine shut off but wanted to hear a voice.
"We copy you down, Eagle," they called, and waited.
"Houston, Tranquillity Base here," Armstrong replied. "The Eagle has landed."
He sounded calm but they had seen his heart rate climb to 156 beats per minute.
Relief winged its way across 390,000km.
"Roger, Tranquillity. We copy you on the ground. You got a bunch of guys about to turn blue. We're breathing again. Thanks a lot."
In Houston it was now 3.18pm, July 20 1969; in eastern Australia, 6.18am, July 21.
- One Giant Leap is a joint initiative with CSIRO, Australia's national science agency, to celebrate the 50th anniversary of the moon landing.