With the Britannia Stadium hosting its first FA Cup fourth round match, the Potters would be hoping to christen it with a win but third division Bournemouth hadn't travelled all this way to lose and wanted to create an upset.
After nine minutes and a sluggish start from both sides, Bournemouth broke down the left flank with Danny Thomas' ball into the middle finally finding Wade Elliott whose shot from 15 yards beat Steve Banks but not the post.
Stoke could have taken the lead in the 21st minute when Bjarni Gudjonsson collected the ball on the right and supplied a cross into a crowded six yard area but James O'Connor failed to direct his header sending it wide of Chris Tardif's left hand post.
The ball finally went into the back of the net just before the break but not from free play, when Tardif received a red card and conceded a penalty for bringing down Brynjar Gunnarsson in the box. Chris Iwelumo was ordered to retake his spot kick but confidently finished past substitute Gareth Stewart.
Tony Pulis' men seemed to have the upper hand at half time after grabbing that late goal, but Bournemouth were giving as good as they got and looked sharp enough to pull it back.
The Potters didn't have to wait long to get their second, as in the 51st minute Chris Greenacre wormed his way towards the left byline and crossed to supply Iwelumo who beat Stewart easily from just six yards.
Stoke clinched the game in the 84th minute when a lay off just outside the box by Greenacre allowed Peter Hoekstra to get inside the area and strike a left footed thunderbolt inside of Stewart's right post.
City manager Tony Pulis seemed pleased with his teams result, commenting: "There were four or five players that were under the weather for today's game, so considering that I am very, very pleased.
"We are hoping for a big tie now, we would love to meet a big club and it would help us financially."