Shels had a two-goal advantage after the first leg finished 3-1 at Tolka Park and they never looked like folding.
First half strikes from Harry Wood and Ali Coote ensured the Dublin side were cruising 2-0 ahead on the night and 5-1 on aggregate.
That's how the tie finished as a sensational Linfield fightback never materialised.
"That was disappointing," said Healy. "We wanted to score the first goal and never gave ourselves that chance. It was a disappointing level of performance.
"All credit to Joey (O'Brien) and his team who fully deserved their victory, and we wish them well.
"That was our most disappointing performance against Shelbourne in the most important of the four games. That is frustrating for me as manager.
"Shelbourne managed the game well, they scored their goals and w