Moura fired Spurs into an early lead when he skipped past Norwich midfielder Billy Gilmour, played a nice one-two with teammate with Son Heung-Min and then produced a fierce strike from outside the box into the top left corner.
Victory lifted Spurs into fifth place with 25 points from 14 matches, 10 points off the top of the table but with a game in hand, as new manager Antonio Conte continued to make an impact.
Norwich dominated possession but could not break Conte's well-drilled defence, with Spurs bright on the counterattack.
Striker Teemu Pukki coming close to snatching Norwich a third-minute lead but hit his shot straight at Spurs goalkeeper Hugo Lloris before Moura opened the scoring on 10 minutes with his laser strike. They were also well-placed to equalise soon after halftime, but Adam Idah spurned a close-range chance with the goal at his mercy after Pukki had miscued an initial effort.
Defender Davinson Sanchez doubled Spurs' deep in the second hald, striking home from close-range after the ball fell to him following a corner. Son sealed a comfortable victory shortly after, slotting home from a layoff from defender Ben Davies.
"I think we have a good mentality. Step by step we work very hard a and try to put on the pitch what Antonio gives us. Our style of play is aggressive now with a lot of intensity. We are doing very well," Moura said.
Defeat for Norwich was the first since Dean Smith took over as manager and left them bottom of the table.
Information from Reuters contributed to this report.