Larne started the brighter of the two sides with captain Tomas Cosgrove causing havoc down the right.
On seven minutes he zipped in a cross from the right and Lusty almost poked it home at the near post, but Cliftonville turned it behind.
Then, six minutes later, the hosts made the breakthrough. Cosgrove again provided from the right, and this time Lusty took the ball on the edge of the area before seeing his deflected effort fly past Morrison.
Cliftonville had to wait until almost the half-hour mark for their first chance. Liam McStravick picked up the ball on the edge of the Larne area and let fly with a low drive that had Rohan Ferguson nervous, but it flashed just wide of the left-hand upright.
Paul O'Neill then went close twice for Larne before Lusty eventually did grab a second on 38 minutes.
The striker then embarked on a mazy run into the Cliftonville area before he shot into the left-hand side of the goal.
Just after the break a mistake at the back almost let McStravick in on goal but his shot was superbly blocked by a lunging Matthew Ridley.
On the hour mark Ridley brought the best of Morrison once more with a powerful header but the goalkeeper made a tremendous reaction save to keep his team.
Morrison almost undid his good work minutes later when he was robbed with the ball at his feet by O'Neill, with the ball rolling to Lusty who shot from 20 yards but Morrison made amends with another superb diving save.
Larne manager Gary Haveron: "The boys worked incredibly, incredibly hard.
"To play the football that they played in the first half, they were excellent. We came out so fast and we got after them nicely.
"We missed some big chances but their keeper pulled off some absolutely unbelievable saves and the only criticism is that we should have been out of sight by half-time."