Marcus Tavernier's smart finish - and an own goal from Malick Thiaw - put the Cherries in control on 35 minutes before the Magpies instantly hit back through Harvey Barnes.
A visibly nervous Bournemouth allowed Newcastle to build momentum in the second period and Joe Willock eventually grabbed an equaliser for the hosts.
That means back-to-back draws for Rose at the start of his tenure in English football and he was frustrated by a familiar story at full-time.
"In the second half, we dropped a bit in everything. Even though we didn't get into real trouble in our box, we conceded again a late goal. This is disappointing, but I'm really happy with the confidence the lads came here today with to try to take the three points.
"At the end, we have to work on the things we need to improve. And that's what we will do in the coming days.
"I try my best. Even the lads. It's not with intention to do it. Sometimes you have phases where things like this happen. We should go for our first win the Premier League to calm down this kind of talks.
"Regarding the points, I'm not happy. We just have two now. It's about results in this level. But performance wise, I see lots of good things."