Palace had failed in their attempts to get the game called off after a coronavirus outbreak in their squad, with manager Patrick Vieira one of those affected by a positive test and absent in north London.
And, after the Premier League ruled the game should go ahead, Moura made matters worse with a virtuoso display, setting up goals for Harry Kane and Son Heung-min either side of his own headed effort in the first half.
Wilfried Zaha compounded his side’s woes with a needless red card before the break, leaving Palace to wish their request had been granted.
Kane’s effort saw him equal a Premier League record for Boxing Day goals, level on nine with Robbie Fowler, and his return to form is another step in the right direction for Spurs.
They remain unbeaten in six Premier League games under Antonio Conte and are within striking distance of the top four with games in hand.
Given Palace’s request to get the game postponed, eyebrows were raised when they made just one change to the side that drew with Southampton 10 days ago, though they were only able to name seven substitutes.
They started the game well as Jordan Ayew fired a shot off target inside the opening 60 seconds from the edge of the area.
But Tottenham began to take control and fashioned some good chances, with Son testing Jack Butland from range, Oliver Skipp skewing wide and Moura glancing a header off-target from Sergio Reguilon’s cross.
A decisive five-minute period swung the game in Spurs’ favour as they scored two quickfire goals and then saw Palace reduced to 10 men.
Kane got his customary Boxing Day goal by firing home in the 32nd minute from Moura’s centre after he was set free by Emerson Royal.
The Brazilian pair combined again two minutes later when Emerson’s cross was headed in from close range by Moura to double the lead.
Palace’s capitulation was complete as, 11 minutes after picking up a first yellow card, Zaha foolishly flung his arms at Davinson Sanchez and was sent off by referee Jon Moss.
Spurs were in complete command and Kane could have added to his tally after the break but failed to keep an effort that came from a clever corner routine down.
The England captain soon made way just after the hour with Tuesday’s visit to Southampton in mind, but Spurs improved their position.
Tottenham had to wait until the 74th minute before extending their lead further as Son got on the scoresheet after more good work from Moura.
The Brazilian again found space down the right and his cross was turned home at the near post by Son for his fourth goal in five games.