After a goalless first half at Fir Park on Saturday, Stuart McCall's side moved ahead three minute after the break when Iain Vigurs' long-range piledriver was parried by goalkeeper Mark Brown, but there was nothing he could do to prevent James McFadden's follow-up finding the net.
That lead was doubled on the hour mark, this time John Sutton on hand to put home the rebound after Brown had not dealt with Craig Moore's initial effort.
Motherwell's lead was cut to just one goal in the 82nd minute, when Yann Songo'o tapped home Graham Carey's corner at the front post, but County were unable to force a late equaliser.
The victory means that Motherwell remain in third place in the Scottish Premiership table and have a seven-point cushion over Dundee United in fourth.
Meanwhile, Inverness Caledonian Thistle remain in the hunt for Europe themselves thanks to a 1-0 triumph over Partick Thistle at Caledonian Stadium.
Billy McKay headed home Ross Draper's cross in the 62nd minute, meaning John Hughes' side - who sit fifth - are nine points adrift of Motherwell, but do have the advantage of a game in hand.
Celtic took another step closer to a seemingly inevitable third straight league title with a comfortable 3-0 home win over St Mirren.
Stefan Johansen's bullet header put Celtic ahead on the stroke of half-time, before Leigh Griffiths doubled that advantage shortly after the hour mark and Anthony Stokes' drilled shot in the 90th minute completed the scoring.
Aberdeen remain in second above Motherwell on goal difference after coming from behind to defeat Kilmarnock 2-1 at Pittodrie.
Kris Boyd put the visitors ahead after 11 minutes, but that was soon cancelled out by Adam Rooney's leveller. Ryan Jack then completed the turnaround in the 72nd minute.
In the day's other game, St Johnstone triumphed 2-0 at home to 10-man Hibernian, thanks to Steven MacLean's strike and an own goal from Ryan McGivern, which came before Danny Haynes' dismissal after 56 minutes.
Hibs have now gone six league games without a win but next up for Terry Butcher's men is the Edinburgh derby with Hearts, where a win could potentially relegate their fierce rivals.
