Further goals from Frazer Wright, Liam Kelly and Mohamadou Sissoko gave Mixu Paatelainen his first victory as Kilmarnock manager as the SPL side avoided any repeat of the 2002 cup upset against the New Broomfield side.
Ryan McCord had given Airdrie hope with a 17th-minute equaliser, while team-mate Euen Grant reduced the deficit with a late free-kick before Sammon netted his third in stoppage time.
Paatelainen made four changes from Sunday's league defeat by Motherwell, starting with Sammon, Kelly and Garry Hay and welcoming Tim Clancy back from suspension.
His side took just four minutes to forge a lead over their Second Division opponents with a well-worked move which saw David Silva play the ball from midfield to Kelly, and he slipped a precise pass to Sammon which the striker slotted confidently low into the net.
Sammon then almost set up a second four minutes later when he pounced on a loose ball at the halfway line and lofted it to Garry Hay, but his close-range header was well saved by Airdrie goalkeeper Mark Ridgers.
The visitors netted a welcome equaliser on 17 minutes.
A dangerously deflected Rhys Devlin shot from outside the box earned the visitors a corner, which was delivered from the right by Philip Johnston and cleared only as far as McCord, who blasted a powerful effort into the bottom right corner.
The Airdrie scorer maintained parity moments later by clearing Frazer Wright's header from a corner off the line at the far post, while Kelly almost put Kilmarnock back in front just before the half-hour with a long-range effort from the right touchline which a stretching Ridgers managed to keep out by holding just under his crossbar.
However, Kilmarnock succeeded in making it 2-1 after 31 minutes when the ball broke to Wright after Sammon and Sissoko had both tried to find space to turn in a Silva cross, and the centre-half fired high into the net.
Kelly had the first effort of the second half, hitting a rising shot from outside the box three minutes after the restart which Ridgers had to beat away for a corner, and the same fate met Watt's attempted curling shot from outside the box.
Kilmarnock made it 3-1 with another impressive goal after 55 minutes, when Sammon ran on to Hay's through ball and left Ridgers with no chance of stopping his confident strike into the bottom right corner.
Kelly made it four for the hosts with the best goal of the evening after 64 minutes, Tim Clancy crossing in from the right after interplay between Sammon and Hay had orchestrated a good move upfield, and Kelly smashed an unstoppable right-foot volley high into the net.
New signing Sissoko scored his first goal for the club 13 minutes later, bulleting home a header from Kelly's cross, but Euen Grant reduced the deficit to 5-2 three minutes from time, his free-kick from outside the box landing in the bottom left corner, and Devlin hit the bar for the visitors in stoppage time.
Sammon completed his hat-trick in stoppage-time by heading home at the far post when substitute Medhi Taouil whipped in an acute-angled cross from the right byline.