The 28-year-old centre-half has been in good form since his summer move from Wigan, having linked up with former Latics boss Steve Bruce once more.
Sunderland managed a 1-0 win over big-spending Manchester City last weekend, a game in which Bramble was particularly impressive.
Having lost out to West Brom the week before, Bramble was happy his side showed their true colours against City and says the Wearsiders are capable of qualifying for Europe.
"We have some really good players here and if we can get some better luck with injuries, I don't see why we can't push for a European place," he told the Sunderland Echo.
"Of course, getting a win over a team like Manchester City helps give everyone a lift, the sort of performance and result which boosts confidence.
"We have a platform to build on now and hopefully we can go to Wigan after the international break and get something from that game as well.
"I know the away form was a problem last season for Sunderland and I've already been part of one poor performance on the road and I don't want another, especially as it's against the manager's and my former club.
"We were bad against West Brom and it just wasn't good enough really. We have shown how well we can play against Manchester City - now we have to maintain those standards."
Bramble admits he was slightly concerned about the reception he might receive at the Stadium of Light because of his Newcastle past, but says the Sunderland fans have been fantastic since his arrival.
"The fans have been great ever since I got here and I want to repay them for that," he added. "I always said when I came here that it was up to me to prove myself as a Sunderland player out on the pitch and things have gone pretty well for me so far.
"I still think I need a few games to be at my best, but I've settled in well and it has been everything I'd hoped for when I decided I wanted to come here.
"It's an excellent club and I'm really enjoying it. You can talk as much as you want, but fans are going to warm to you if they see you are playing well on a matchday. That's all that matters."