The Staggies had ended Celtic's hopes of a treble with a 3-1 League Cup semi-final win at Hampden Park two weeks ago and the pressure on Hoops boss Ronny Deila increased with a 2-1 defeat at Aberdeen days later.
A 2-0 win over Lowland League side East Kilbride in the Scottish Cup last weekend did little to appease critics of the Norwegian and there was little to heat up the Hoops fans on a bitterly cold day until Griffiths rifled in a drive from 12 yards just before the break.
Defender Dedryck Boyata headed in a second in the 56th minute and the home side ran out comfortable winners, putting pressure on the Dons, who play Inverness on Tuesday morning (AEDT). Celtic, though, will have a game in hand afterwards and a vastly superior goal difference.
Australia attacker Tom Rogic came on for Celtic with 14 minutes remaining, picking up a late yellow card for his troubles.
After pairing Griffiths with new signing Colin Kazim-Richards against East Kilbride last week, Deila returned to using his top scorer as the lone striker, with goalkeeper Craig Gordon, defender Mikael Lustig and midfielders Stefan Johansen, Nir Bitton and Stuart Armstrong returning.
The league leaders forced three early corners to show intent and in the ninth minute Staggies keeper Scott Fox made a great save from Gary Mackay-Steven's drive after County skipper Paul Quinn had conceded possession to Johansen.
Moments later at the other end, Australia international Jackson Irvine - playing against his former club - drove just wide from the edge of the box after Bitton had failed to deal with a rather aimless Richard Foster cross.
Griffiths had a shot blocked by a clutch of dark blue jerseys during a goalmouth melee, but as the first half continued the home support became increasingly agitated at their side's lack of tempo and drive.
And, when Griffiths rounded Fox after latching on to a Johansen pass only to hit a post, it hinted at an afternoon of frustration for the home side.
However, just before the break, Celtic left-back Kieran Tierney crossed into the area and, when Staggies defender Chris Robertson headed clear to Griffiths under pressure from Johansen, the Scotland striker quickly returned the ball low past Fox.
Mackay-Steven blazed a shot over the bar early in the second half after being set up by Johansen in a swift break, but it was no indication that the Hoops had stepped up a gear.
However, if the performance was not sparkling the points were effectively secured when Armstrong's cross from the left found Boyata at the back post and he headed powerfully past Fox.
The two-goal cushion allowed Celtic to relax a bit, invite County on to them and hit them on the break.
Johansen, Erik Sviatchenko and Griffiths had further efforts as the Highlanders left themselves more exposed than they would have liked in an effort to get a foothold in the game.
Five minutes before the end Gordon produced a spectacular one-handed save from Jonathan Franks before Ian McShane fired over from inside the box, the last real action before the match petered out through four minutes of added time.
At the bottom of the table, there was a huge win for Kilmarnock against Motherwell.
Kilmarnock leapfrogged their opponents into 10th place after Greg Kiltie and Craig Slater earned them a 2-0 victory with goals early in the second half.
Scott McDonald played a full game for Motherwell.
Bottom side Dundee United held Hamilton to a goalless draw after Eiji Kawashima saved Ali Crawford's 86th-minute penalty. Hamilton had Michael Devlin sent off in injury-time.
Heart of Midlothian's match against Partick Thistle was postponed after the Tynecastle pitch failed an early-afternoon inspection.
The game, due to begin at 3pm local time, was called off less than two hours before kick-off.
Hearts announced on their official website: "Today's Ladbrokes Premiership match between Hearts and Partick Thistle has been postponed.
"Referee Bobby Madden inspected the pitch moments ago and judged it to be waterlogged, hence the match being called off."