The Eels dominated possession and field position in the opening exchanges of the second half but winger Sean Russell was twice denied tries within a minute as North Queensland's goal line defence held firm.
Referee Todd Smith correctly disallowed a 48th minute try after a quick catch and pass from Will Penisini travelled forward before Russell caught it and strolled over untouched.
Russell again looked set to score from the ensuing scrum, due to a Cowboys knock-on before the forward pass, but he lost the ball attempting to ground it as Ben Hampton came across in cover to tackle him.
Sivo finally broke the 6-6 deadlock when he scored in the 55th minute after Brown, Gutherson and centre Bailey Simonsson combined to create a passage for the Fiji star to the tryline.
Russell put the Eels further ahead when he was finally rewarded with a try in the 64th minute after quick hands by the Parramatta backline.
However, the Cowboys refused to surrender and winger Kyle Feldt scored two late tries - both of which were laid on by Drinkwater.
"What he said before the game I really liked. He wanted to touch the ball and have 20 carries and when he has got his hands on the ball we are a better team," Cowboys coach Todd Payten said of Drinkwater.
"He took some tough yardage carries, he took some good options in good ball and he was a real threat for us. The kid cares about how his footy grows and cares about how we go as a team.
"I have really liked his intent and effort, particularly these last six-to-eight weeks."
Feldt's first try was created with a long pass by Drinkwater on the outside of Sivo, while the second followed a Parramatta knock on after Feldt chipped over the top of his wing opposite.
Mitchell Moses secured the win when he scored in the 79th minute.