Nugent-Hopkins scored at 13:46 to make it 4-1, ending a shutout streak of 153:46 for the Oilers in the series.
Toronto shut out Edmonton 4-0 on Saturday and 3-0 on Monday.
"Obviously we feel really good about the results and how we handled leads and built leads," Maple Leafs coach Sheldon Keefe said. "A lot of really good things came out of this series for us. Coming in, we were playing against a very good team that was as hot or hotter than any team in the NHL and coming with lots of confidence. For us to get results like this is a good and healthy sign."
William Nylander made it 5-1 at 18:22 and Hyman scored on the power play at 10:33 of the third period for the 6-1 final.
Oilers captain Connor McDavid, who leads the NHL with 40 points (14 goals, 26 assists) did not have a point in the series.
"For whatever reason we couldn't figure those guys out and they had us pretty figured out," McDavid said. "We have to move on to the next one, we have a pretty big one against Calgary (Flames), back to the Battle of Alberta (on Saturday) and then three against an Ottawa (Senators) team that is playing very, very well. This is a big point in our season for us."