Following an indifferent start to the season, Blues have now gained plenty of momentum with the weekend's 4-1 Championship victory over Sheffield Wednesday followed by a 3-1 Capital One Cup success over holders Swansea.
Jesse Lingard, unavailable against the Swans due to the terms of his loan from Manchester United, provided a breath of fresh air as he became the first player in over 60 years to mark his Blues debut with a hat-trick as he netted all four goals against the Owls.
Centre-back Burn and midfielder Adeyemi then netted their maiden goals for Lee Clark's side to help down Swansea - Matt Green chipping in with the other - and the pair both believe everything is finally coming together for the Midlands club.
Speaking after the cup success, Burn said: "We've played really well all season, taking away the Burnley game (a 3-0 defeat), but now we're starting to get results you can see the confidence is starting to build around the team and lads are more confident on the ball.
"Different players are starting to score, Matt Green has got a few goals now and he looked an absolutely great player here. So it's nice we're playing with confidence and that always helps.
"Jesse came in and did really well on Saturday and showed the lads that we could take our chances and we're going to get results, and we've done it again."
Adeyemi added: "I think prior to these two results we were still confident because our level of performances have been reasonably good.
"We'll take a lot of confidence out of the fact that we've scored seven goals in two games because that was one area in which we were struggling a little bit.
"Jesse's probably been the catalyst for that before then moving on to this win."