Gary Hooper's goal after 30 minutes proved enough to give the Canaries a much-needed victory, which earned Hughton some breathing space following seven Premier League defeats before Saturday afternoon.
Palace, boosted by last weekend's away win at Hull, were probably worth a point for their second-half efforts, but Norwich eventually closed out victory in front of a new all-seater record 26,851 attendance at Carrow Road.
It was a bright opening by Norwich, whose inconsistent form had seen Hughton's position come under scrutiny following a £20million summer spending spree.
Two of those new arrivals combined to create the first chance on nine minutes.
Teenager Nathan Redmond, signed from Birmingham, clipped in a free-kick which Sweden forward Johan Elmander, on loan from Galatasaray, crashed against the crossbar, with the ball hitting the line then bouncing clear and no signal coming from the new Goal Decision System.
Moments later England Under-21 winger Redmond, with hopes of maybe forcing his way into World Cup contention, drilled the ball goalwards after cutting in from the right, but Palace keeper Julian Speroni saved at the near post.
At the other end, Cameron Jerome appealed for a penalty when his shot cannoned off Norwich captain Sebastien Bassong's arm, but referee Chris Foy was not interested.
The Canaries finally went ahead on the half-hour.
Redmond was the instigator with a determined run down the right before Elmander swept the ball into the penalty area.
It dropped to Wes Hoolahan at the far post and he squared it for Hooper to roll the ball into the net.
Palace, though, almost equalised straight away.
Jason Puncheon's cross from the right was palmed out by John Ruddy and the ball dropped to Barry Bannan.
The midfielder cracked a fierce drive goalwards, which was headed for the top right corner before Martin Olsson somehow headed it on to the crossbar and away.
Norwich returned to a spell of possession as half-time approached, with Redmond again the main outlet down the right as he set up Elmander at the near post, but the Swede shot wide.
A lovely piece of control from Hooper and back-heel to Elmander saw Redmond away again at the start of the second half, but after cutting inside from the left, the winger's shot was fired high into the River End.
Redmond curled a ball across the six-yard box which would only have needed a touch to put Norwich 2-0 ahead as the hour approached.
However, the Eagles soon pushed Norwich back again, with Dean Moxey's low cross from the left somehow hacked over his own crossbar by Bassong just two yards out.
Palace substitute Jimmy Kebe sent a shot on the turn just wide and, despite an anxious four minutes of added time, Norwich held on to put some distance between them and the relegation zone.