Wrexham played some superb football particularly in the first half - but missed several great chances of taking City to the cleaners.
Hector Sam had a real off day missing at least three golden chances, one when he was put through in the first half by Lee Jones but he hit his shot over the bar.
Jones also missed a great chance to break the deadlock midway through the second half when Darren Ferguson put him through but the livewire striker hit his shot past the far post.
City had had an early chance which Lee Miller fired wide but in the 75th minute the visitors came close to winning when Tony Butler's header from a Luke Wilkshire corner hit the bar.
Wrexham dominated the game in the first half - Sam failed to connect with a Carlos Edwards cross and then saw a deflected shot go into the arms of keeper Steve Phillips.
Wrexham suffered an early blow when skipper Brian Carey, who had a midweek virus, had to leave the field and was replaced by up and coming youngster Craig Morgan.
Just before half time when City were on the back foot, Louis Carey did well to get in a block tackle on Wrexham's Chris Llewellyn in front of goal to stop him connecting with a Lee Jones cross.
Early in the second half, Phillips did well to beat out an Edwards shot.
After the match Wrexham manager Denis Smith said: "It was disappointing not to get a win, especially after dominating the first half.
"They packed the midfield in the second half and the game petered out. Before the season started you would settle to get a point from City, but we are disappointed."Robins boss Danny Wilson said: "It was not a classic by any stretch of the imagination but it was a game of limited chances.
"We looked solid in the second half and looked like a side that was going to make it."