Big keeper Clayton Ince earned Crewe a replay with a breathtaking late save.
Ince took off to his right to push away a goalbound effort from winger Wade Elliott as Bournemouth pressed for a winner.
If Ince was Crewe's hero, his opposite number Chris Tardif was Bournemouth's.
Tardif, who has twice been taken to hospital this season with facial injuries, kept out efforts in the second half from John Miles and David Vaughan.
He reserved his best for the 69th minute when he arched backwards to push a corner from Kenny Lunt on to the bar.
Crewe, winners of six of their last seven matches, struggled to find their best form against Bournemouth who are a division below them.
Crewe went in to the match without injured trio Rob Hulse, Dean Ashton and Rodney Jack, who have scored 40 goals between them this season.
Bournemouth have only themselves to blame for needing to go to the replay at Gresty Road.
They dominated the first half and both Steve Fletcher and Elliott should have done better with clear openings.
Crewe manager Dario Gradi was far from delighted by the performance or the result.
"I am happy to be in the hat for the fourth round especially when it looked like we would lose," he admitted after the match.
"I do not want another epic cup run, strange as that may sound.
"It cost us relegation last season and I do not want it to cost us promotion this year."Bournemouth manager Sean O'Driscoll said: "We could have done without the replay and we had the chances to win.
"I was happy with our performance but not with the result. We have only a small squad and an extra match will test our resources still further."