Along with Cole’s double, Ryan Bell also netted as three second half goals saw Golds to a home 3-0 second-round triumph over division-lower Southern Combination League Division 1 side Storrington.
Strong winds and heavy rain made for horrendous conditions but Littlehampton managed to come through the tie courtesy of a strong showing after the break.
Swans managed to withstand what Golds threw at them in the first half but an early goal after the interval broke their resistance.
Bell got the opener and a double from substitute Cole soon followed.
After being introduced from the bench, Cole added a second, before rounding off the scoring to ensure Littlehampton’s progression.
Golds wasted a three-goal advantage as they drew 3-3 at Eastbourne United in the Southern Combination League Premier Division on Saturday.
Lucas Pattenden, Alex Laing and Mark Zydonik all struck in the opening half-hour to put Littlehampton in a commanding position.
United stormed back in the second half, though, with efforts from Sam Davison, Callum Hart and Sam Hart ensured the home side foughtback from 3-0 down to take a point.
Sam Hart’s equaliser came five minutes into second half stoppage-time, which Golds boss Ady Baker admitted was tough to take.
He said: “The lads gave it everything but having been 3-0 up we should have come away with the victory.
“If you score three times, especially away from home, you’d certainly hope it would be enough to secure victory.
“Everything seems to be going against at present, all we can do is keep working hard to turn things around and hopefully results will come.”
The point lifted Golds off the bottom, up a place above rivals Arundel on goal difference.
Littlehampton will be hoping to continue their climb up the table when they travel to Loxwood in the league on Saturday.
LITTLEHAMPTON TOWN v Eastbourne United: Hutchings; Sharp, Farrell, Zydonik, Chaplin; Kempson, Bell, Bankole; Heath, Laing, Pattenden. Subs: McKay, Janman, M.Bromage.
LITTLEHAMPTON TOWN v Storrington: M.Bromage; Sharp, Bell, Janman, McKay, Chaplin, Pattenden, Noble, Laing, Gray, Heath. Subs: Cole, Hutchings.