The Ballymacarn Park club are chasing a hat-trick of domestic league titles and scored seven tries in the bonus point victory over the County Tyrone team.
Dungannon made life difficult for Hinch in the first half.
After a scoreless opening quarter, the visitors took the lead with a Ben McCaughey penalty on 22 minutes.
Hinch eventually made a breakthrough on the stoke of half-time with hooker Clayton Milligan going over for the game’s opening try – Conor Rankin converted to give the home side a 7-3 advantage at the break.
Hinch scored three tries in six minutes after the restart.
Openside flanker Zac McCall went over in the 41st minute, Milligan went over two minutes later before completing his hat-trick in the 46th minute with Rankin converting all three.
Replacement Arthur Johnston was next to cross for Hinch on 51 minutes with Rankin again adding the extras.
The side’s exchanged two tries each in the final quarter.
Armagh were pushed all the way by Rainey at Hatrick Park in Division One Group B, but Chris Parker’s side emerged with a 36-31 bonus point win.
Centre Timmy McNiece got the first two of Armagh’s six tries.
Winger Glen Faloon also grabbed a brace with hooker Jack Treanor and centre Matthew Hooks each crossing once.
Ronan McCusker and Tommy O’Hagan were among the Rainey scorers as the Old Boys took two points from the game.
Malone also picked up a bonus point victory in Group B as they beat Ballymena 38-18 at Gibson Park, where Darryl Maxwell and Andy Bryans were among the try scorers for the home team.
Clogher Valley’s first game in senior rugby ended in a bonus point victory as they beat Bangor 31-12 at Upritchard Park.
The County Tyrone side outscored Bangor by five tries to two in the Division Two clash.
Callum Smyton got two of Clogher’s tries with Andy Dunwoody, Rodney Bennett and David Stinson all crossing while former Ulster winger Craig Gilroy got over for Bangor.