Miguel Almiron's goal saw him end a run of 2190 minutes and 46 shots without scoring in the Premier League.
Newcastle welcomed back Almiron and Jonjo Shelvey after they missed last Saturday's 1-0 defeat at Burnley with minor injuries and they provided a welcome boost as the Magpies remained unbeaten at home since the opening day of the season.
The midfield pair had been instrumental to their team's run of seven points from nine before the Turf Moor setback and added some guile to go with Newcastle's grit.
Both showed some neat touches in midfield while Almiron curled a tame effort straight at Palace goalkeeper early on while Shelvey sent a speculative effort just over from near the halfway line.
Then with time running out, Almiron found himself in space in the box to pounce on Carroll's knock-down and his relief was clear, the 25-year-old peeling off his shirt in celebration, which saw him receive a booking.