Haveron's side broke the deadlock in the 51st minute when Dungannon failed to clear Ryan's delivery from the left wing with Randall able to control the loose ball and slot home from close range.
Shortly after, a James Knowles free-kick pinged off the outside of the post and Ryan then saw his low effort denied by the left-hand post as both teams upped the tempo.
However, that was short-lived as the game returned to a slower pace like in the first half, with McGovern's strike easily thwarted by Rohan Ferguson.
Dungannon continued to push for an equaliser, but the Larne defence were able to clear any danger created.
Adam Glenny had a huge chance with the last kick of the game when his strike from 25-yards flashed across the face of goal, to the frustration of the home fans.