The former Ross County forward scored twice in a four-minute spell early in the second half – his second from a free-kick – to give the Highland League side control.
Scott Barbour extended Formartine’s lead after 69 minutes and three minutes later, Gethins completed his hat-trick to set up a fourth-round tie at Partick Thistle.
Morton had no such problems with a 6-0 win at Junior side Beith to secure a clash with Falkirk in the next round.
Ross Forbes put them ahead after 11 minutes with a trade-mark free-kick to ease any pressure on Jim Duffy’s side.
Michael Tidser made it 2-0 six minutes afterwards and Jamie McDonagh added a third three minutes from the break after a fine solo run.
Thomas O’Ware netted the fourth six minutes into the second half with a low shot which went in off the inside of the post.
Jamie Lindsay also managed to get on the scoresheet with 12 minutes left, netting with a left-foot shot.
Central defender O’Ware, Morton’s top scorer, completed the scoring in the final minute with a downwards header.
Stirling Albion had to work hard to beat Wick Academy 2-0 at home. In an evenly matched affair, Blair Henderson broke the deadlock two minutes before half-time then Dylan Bikey added the clincher with six minutes left to book Stirling’s fourth-round place.
A hat-trick from Colin McMenamin gave Stenhousemuir a 3-1 replay win over Forres Mechanics at Ochilview and gain them a fourth-round tie against St Johnstone. Liam Baxter pulled back a late consolation of the Can Cans while Stenhousemuir finished with ten men when David Marsh was sent off.
Stranraer squeezed through 2-1 at home to East Kilbride due to an own goal from Adam Strachan with just a minute left.
Sean Winter gave East Kilbride a shock lead after 14 minutes but Stranraer were level after 26 minutes when Willie Gibson blasted a penalty high into the net. Then came Strachan’s unhappy ending and Stranraer now travel to Aberdeen in round four.