The 21-year-old capitalised on the Bees' inability to clear the ball, firing past Daniel Bentley early in the second half.
It was his first goal since joining the Championship club on loan from Chelsea. And speaking to the Lancashire Telegraph after the game, Palmer said a missed chance earlier in the season stuck with him on Saturday.
He said: "I had a similar chance at Ipswich and I remember thinking that I just had to get this one on target. I got the power on the ball and thankfully it went in.
"If I scored and we didn't win then the goal wouldn't have met anything to me, so to score and win the game made it a great day.
"We won't get ahead of ourselves and we'll keep on going, but we're confident in the dressing room that we can be unpredictable and achieve things that people don't think we can."