The Magpies came from behind to snatch a 2-2 stalemate at St James' Park, with Atsu spurning an opportunity in the 78th minute to get on the scoresheet.
The Ghanaian made his first start and lasted the entire duration since returning from the recent Africa Cup of Nations in Gabon.
“I’m very happy when I start," Atsu told his club's official website.
"I always try to win every game because it’s very important for the team and also for me.
"But for me personally, it’s a disappointing result, because I could have scored and we could have also won the game."
Aaron Wilbraham and David Cotterill gave Bristol a two-goal lead in the first half before Korey Smith (own goal) and Ciaran Clark brought Newcastle back into the game.
“I think we deserved to win the game," Atsu said.
"We created a lot of chances. In the first half, we conceded two goals, and I think we did very well to fight back to draw 2-2. But I thought we deserved to win the game even though we were two down – we came back and still we were creating enough chances.
“Sometimes, it happens. We tried to give our best to win the game, and I think they were also lucky – they could have conceded a third goal.
"We have to take our chances."
On a season-long loan from Chelsea, Atsu has made 20 league appearances thus far, scoring thrice.
Newcastle sit one point adrift of first-placed Brighton and Hove Albion, ahead of their top-of-the-table clash on Tuesday.