Alex Scott had given the home side a deserved lead just before half-time, finishing coolly after being teed up by Evanilson, and Bournemouth looked to be closing in on victory until Tarkowski pounced on a mix-up at a corner in the second minute of stoppage time to snatch a point for David Moyes' side.
Moyes admitted his side had ridden their luck.
"I didn't think a goal was coming. We stuck at it, resilient, we were good away from home last season so hopefully we can keep it going," he said. "We needed somebody to come up with something. We did some amazing blocks late in the game. Pleased with a point."
Asked about the transfer window, which closes on Tuesday, Moyes added: "Might want the transfer window left open so we can get a few more players in! Let's see what happens.
"I have said we could do with strengthening and hopefully we can do that. We have a good basis, let's hope we can build on it. I don't think we can afford to lose any players, certainly not our better ones."
Rose was left to reflect on a missed opportunity for a first win in charge.
"It was mostly about the result but as a coach you evaluate the performance as well and in most phases of the game it was really good," he said. "It was quite an open game, quite up and down. Today for me it was not even a set piece for the goal because we had already cleared the situation, but I don't want to blame anyone because I saw so many positive things.
"I am really thankful for the warm welcome and I really appreciate being coach of this team. Like last week we would like to present our fans with a win but we will have to wait one more week."
Goalscorer Alex Scott felt Bournemouth deserved more.
"I think we deserved to win the game. A mistake at the end and it has cost us three points," he said. "We had chances to go and win the game and didn't take them so for now it is head down and focus on Newcastle next week."
On his goal, Scott added: "Evanilson is really good at running in behind and holding on to the ball, and for me as a number eight I want to score more goals so making that extra burst to get on the scoresheet is important."
Scott also reflected on settling in under Rose after Andoni Iraola's departure for Liverpool.
"It's similar, probably a few more patterns on the ball. It's high-pressing football, exciting football for the fans. We're learning, he's a top manager and he has got top coaching staff as well, so we have really enjoyed the weeks he has been here."