The Reds were last beaten in the league at the City Ground in September last year, a record stretching back 29 games, and a Crystal Palace side who had lost their previous four road trips never threatened to upset the form book.
Centre-half Luke Chambers opened the scoring with a first-half header and Tudgay, making his first appearance since joining on loan from Sheffield Wednesday, grabbed the second moments after the break.
Garath McCleary completed the scoring late on as Forest leapt back into the top half of the Championship and left the Eagles in the bottom three.
The hosts set about Palace from the first whistle and livewire midfielder Lewis McGugan was twice denied by visiting keeper Julian Speroni in the opening stages, the Argentinian parrying his long-range snap-shot before beating out a fierce angled drive.
Rob Earnshaw also fired over and had the ball pinched off his toes by Claude Davis as he was about to meet Chris Gunter's cross.
Forest thought they had taken the lead after half an hour when Tudgay played in Earnshaw but the Welshman had bundled Speroni, as well as the ball, into the net.
However, within two minutes they were ahead when Chambers rose highest to meet McGugan's corner with a firm header for his first goal of the season.
With Palace on the ropes, Earnshaw could have immediately doubled the lead when he ran onto Radoslaw Majewski's through-ball but Speroni was quickly out of his goal and blocked the shot.
Despite the constant pressure the Eagles held out until half-time, only to press the self-destruct button 35 seconds after the restart.
Tudgay was allowed to latch onto a long clearance from Guy Moussi and Speroni found himself in no man's land as the striker lobbed the ball over him and into the net.
Forest could have had more but for Speroni, who smothered Earnshaw's effort on the line and pulled off a stunning one-handed save from Chambers' volley.
From the resulting corner Chambers planted a header narrowly wide before Speroni came to Palace's rescue again by beating away Paul Anderson's shot.
Forest, in cruise control, were able to hand another late run-out to on-loan Arsenal midfielder Aaron Ramsey as he builds up his fitness following his broken leg.
A cute backheel from the Welsh youngster almost set up Nathan Tyson but his shot was deflected wide.
But the third goal finally came when Tyson teed up his fellow substitute McCleary to fire home six minutes from the end.