The Eagles' in-form frontmen made it three wins out of three for new boss Ian Holloway and stretched their unbeaten run to 14 matches.
Murray grabbed his sixth and seventh goals in four games, and Brazilian Moritz - whose inclusion from the start was the first change Holloway has made to the team since he took over - rifled in his third in three.
Derby, by contrast, failed to muster an effort on target and crashed to a second defeat in a week in south London following last Saturday's loss at Millwall.
And the Rams were architects of their own downfall when they gifted Murray a simple opener after 12 minutes.
Yannick Bolasie swung in a corner and the striker was left unmarked at the far post for an easy header.
They could have doubled their lead eight minutes later when Derby's defending from corner's was exposed again. Moritz's cross was met by Palace skipper Mile Jedinak, but this time the header flashed wide.
And midway through the first half another Moritz corner caused problems with Richard Keogh almost diverting the ball into his own net.
After the break Wilfried Zaha, fresh from his brief England debut in Sweden, showed a glimpse of his class with a trademark run and cross to Moritz, who should have doubled Palace's lead but scooped his shot over from eight yards.
Palace went 2-0 ahead when Damien Delaney's ball into the area was headed down by Murray and there was Moritz to apply the finish with a sweet left-footed volley.
Dikgacoi came close with a near-post effort from Moritz's corner but his shot flew inches over, but Murray wrapped up another Palace victory with a strong run and shot into the roof of the net with nine minutes left.