In a quiet opening, the first chance fell to Linfield as Adam Frizzell's cross from the left found Kirk Millar at the back post but the winger was too far from goal to get enough power in his header to trouble Declan Dunne.
Dungannon hit the woodwork with their first attack when Corey Smith cut in from the left flank and his shot bounced once and clipped the outside of the post with goalkeeper Chris Johns appearing to have the danger covered.
Linfield took the lead in the 14th minute when Kyle McClean's pass sent Frizzell clear in the left-hand channel and his low cross was converted from close range by the in-rushing Fitzpatrick.
The home side almost doubled their advantage inside the opening 20 minutes when Isaac Baird dispossessed Gael Bigirimana in midfield before playing in Fitzpatrick, but Swifts goalkeeper Dunne dived to his right to push the striker's effort away.
Linfield had the ball in the net again before the half hour mark but Darragh Leahy's strike was ruled out for a foul by Baird on Paul Doyle earlier in the move.
Dunne then produced a vital double save just before the interval, first denying Millar's shot and then getting up quickly to thwart Fitzpatrick's follow-up effort.
Dungannon had an early opening in the second period as Bigirimana set up Leo Alves, but he could only poke a left-footed effort wide of the post.
It was Linfield's turn to hit the framework of the goal eight minutes into the second period as Millar's cross from the right found McClean but he could only force his shot against the inside of the near post.
As the weather conditions worsened, Fitzpatrick then saw a shot pushed away by Dunne as Linfield chased a decisive second goal.
It duly arrived on 64 minutes when Millar played a through ball which Dungannon defender Danny Wallace was unable to cut out, leaving Fitzpatrick to race clear before knocking the ball through the legs of the advancing Dunne and then rolling the ball into the unguarded net.
Fitzpatrick completed his hat-trick with 20 minutes remaining as he converted from close range from Ethan McGee's low cross from the right wing.
Dungannon pulled a consolation goal back with three minutes left when Kobei Moore latched on to James Knowles' through ball to force a shot home but it was never going to affect the outcome of the game.