Bournemouth went second in the table and stretched their unbeaten run to six games with a point at lowly Orient - but with steadier second-half finishing they could have won the match.
Cherries substitute James Hayter twice could have been cast as the hero after coming on for injured top scorer Alan Connell.
In the 72nd minute he saw his free header from eight yards saved by Orient keeper Glenn Morris - and then six minutes later he was denied again by the teenage keeper.
But it was not all one-way traffic. Orient substitute Jabo Ibehre found himself in the clear in the 64th minute but the chance evaporated when he delayed his shot for too long.
The home side might have had a penalty when Liam Ridgewell appeared to push Orient skipper Dean Smith in the penalty area late on.
After the game Bournemouth manager Sean O'Driscoll said: "I thought we had the best two chances of the second half and I was disappointed that we did not score one of them.
"But I was delighted to keep a clean sheet - I knew this would be a difficult place to come to."
Meanwhile Leyton Orient manager Paul Brush said: "They're second in the league and we've not been outplayed but although we defended well we just didn't create enough chances and that's the most disappointing thing."
It was the visitors who started well and should have scored in the second minute when Danny Thomas found space down the right and fired across the goal when he should have hit the target.
But both sides had their moments as Matthew Lockwood saw a free kick tipped away by Bournemouth keeper Neil Moss.
Carl Hutchings crashed a 20-yard effort over the bar and Kevin Nugent headed over from close range in a frantic first half.
Despite the lack of goals there was plenty of excitement for Orient's biggest crowd of the season as Bournemouth's Brian Stock saw a second-half corner hit the post and Morris also saved well from Connell before he limped out of the action