Orient picked up only their second away point of the season, but missed several chances to win the game.
Home goalkeeper Andy Dibble was voted man of the match after a string of saves and Wrexham had to survive three injury-time corners to hang-on for a draw.
Orient manager Paul Brush said: "We did well enough to win the game. We had plenty of spirit and if we can get two wins together we can improve things. We are not taking our chances but that's the edginess you get if you are not scoring enough goals."
O's keeper Glenn Morris saved an early shot from Lee Jones but was in trouble later when he raced off his line to challenge top scorer Andy Morrell.
He was stranded off his line as defenders scrambled the ball away for a corner. Another escape followed when Jones pulled the ball back across goal, and Carl Hutchings cleared as Morrell raced in.
Morris saved again from Darren Ferguson and at the other end Dibble made a great save to keep out a Kevin Nugent header from a pinpoint cross by the hardworking Matthew Brazier.
Lee Thorpe wasted a chance by shooting straight at the keeper and after that Dan Bennett put the ball through to substitute Hector Sam, who rounded Morris and put his angled shot into the side netting.
Hutchins missed a sitter when he should have sealed it, heading wide with only the keeper to beat.
Wrexham manager Denis Smith said: "It was a bad day at the office. People thought we would have turned Orient over, having scored in all our league games but they were one of the pre-season favourites."