Slater, who has been associated with United for 26 years as player, captain and manager, asked to leave the Moor Park role due to work commitments within his day job as a manager at Tarmac.
Lanark president George Reid told the Carluke and Lanark Gazette: “Colin felt that he wasn’t doing himself justice and he wasn’t doing the club justice.
"To do what’s best for the club, Colin said if there was somebody else we could bring in, we should go and do it.”
Up until last week, new United co boss Eeles, 39, was still registered as a player at Blantyre Victoria, for whom he played on the right wing.
He has been involved in football for 20 years, also playing at Irvine Meadow, Glenafton, Kilbirnie and Hurlford.
"Colin met Simon last week and more or less sold our club to him,” Reid added.
"One of the main things you need to run a club like ours is you’ve got to have good contacts for players.
"So to have a guy who was in the twilight of his career, still playing at 39, he knows current players that would come and help him out.
"One of the biggest pressures of the Lanark job right now is getting players in.
"You can strip seven subs now but we’ve never needed our seven subs.”
The other co boss, Soutar, does have coaching experience as he moves to Lanark having run Blantyre Vics under-20s.
Over the last 10 years, he has coached at pro youth level at a variety of clubs including Inverness Caledonian Thistle, Ross County, Greenock Morton and Queen’s Park.
Soutar also spent two years coaching in the USA, in Seattle and Washington, and holds his UEFA A Licence.
A United quote on behalf of the two new gaffers read: “We are both extremely honoured to take over at Lanark, excited to meet the players and get started.”
Eeles and Soutar have a big job on their hands, with Lanark bottom of West of Scotland League Conference C with eight points from 18 games after Saturday’s 2-0 home defeat by St Roch’s.
This Saturday, Lanark are playing host to 10th placed Yoker Athletic in the league, KO 1.30pm.