Sam Allardyce's team failed to find the back of the net at fortress Reebok, despite having the lion's share of the chances and dominating the play.
Amazingly, Bolton had 11 shots off target and also hit the target on six occasions.
But some superb saves by Hammers keeper Shaka Hislop and some poor finishing meant this game will go to a replay.
As early as the seventh minute Kevin Davies saw a shot blocked by Danny Gabbidon, before West Ham responded with a break at pace at the other end.
New signing Dean Ashton fired wide with a fierce left-footed volley from 25 yards, while at the other end, man of the match and Bolton's inspirational captain Kevin Nolan forced an excellent close-range save from Hislop.
A minute later big defender Radhi Jaidi saw a header saved by Hislop as Bolton pressed for an opener and a quarter-final berth.
The impressive Nolan volleyed wide in the 29th minute as Bolton kept West Ham at bay, before going close again three minutes later, only for Ashton, chasing back well, to deny him with an excellent block in his own penalty area.
West Ham finally started to click seven minutes before the interval and Paul Konchesky saw a shot blocked by Tal Ben-Haim.
Veteran midfielder Gary Speed forced Hislop to make an excellent save from a powerful header in the 42nd minute, before Nolan again tested Hislop a minute before the interval, only for the on-song keeper to save again.
It was Nolan who led the charge in the second half and in the 54th minute he shot wide and then two minutes later Davies did well to clear a shot from Lionel Scaloni.
In the 57th minute Nigerian international midfielder Jay-Jay Okocha shot wide as Bolton pressed and pressed for a deserved opener.
A minute later Nolan fired wide and then Davies, 21 minutes before the final whistle, forced the best stop of the game from Hislop.
Allardyce introduced substitute Ricardo Vaz Te late into the action and the Portuguese striker nearly had the desired effect giving his team a place in the next round.
But Scaloni, this time playing in his preferred defensive role, charged down the striker's efforts and West Ham survived to live another day.
The replay is bound to be an interesting affair with West Ham now slight favourites to edge through with home advantage on their side.