Eddie Howe's side registered 23 shots during the 90 minutes against Patrick Vieira's Eagles, with Miguel Almirón's deflected effort clipping the post in the first-half at St. James' Park.
The hosts thought they had taken the lead through Tyrick Mitchell's own goal but the goal was disallowed in controversial fashion after a VAR check, with Joe Willock adjudged to foul Palace goalkeeper Vicente Guaita by referee Michael Salisbury.
Anderson, introduced from the substitutes' bench in the closing stages for his first competitive senior game at St. James' Park, told nufc.co.uk: "Watching from the side, I thought we did really well and created so many chances whilst keeping the ball out of our own net. I think, to get to that next level, we need to put our chances away to start getting wins on the board.
"We've been very unlucky today, to be honest. We've been in the right areas and places but none of the chances seemed to hit the back of the net. For me, I'm trying to get that first goal and hoping it comes soon so I can start kicking on from there.
"I'm confident in the lads to get three points in the near future. You can see when we've picked up draws, everyone is fighting very hard for the win and I think that victory will come soon.
"The defending has been very good and we just need to be a bit more clinical and have that cutting edge when taking our chances in front of goal."
The 19-year-old, who made his Premier League debut at the age of 18 away at Arsenal in January 2021, expressed his delight in playing his first English top-flight game for his boyhood club at St. James' Park.
A representative of Scotland at under-21 level, the highly-rated youngster vows to continue building on his recent appearances at first-team level.
The teenage midfielder, who joined the Magpies' Little Benton academy at the age of eight, added: "I loved it! I try to enjoy every minute I am out there as it hasn't happened often and I hope to be out there more.
"It's different playing away from home," Anderson added. "I've dreamt of playing at St. James' Park in the Premier League for a while now. When you get onto the pitch, you sort of focus into the game and cancel it out. Afterwards, you realise it's quite amazing when you're looking around the stadium.
"I've been trying to earn the gaffer's trust in putting me on and hoping I can make that difference. I thought I played well but I think I was just missing that goal or assist to try and change the game."
