Meekings bundled over the line in the dying embers of the game to give Caley Thistle a 2-1 win over St Johnstone, who had looked like they would snatch a point.
Ross Draper gave Inverness the lead but a long-range effort from Richard Foster pulled the visitors level with 10 minutes to go.
Just when it looked like Inverness could not barge the door down, Meekings' close-range finish sparked wild scenes of celebration.
The home side made three changes from the side that lost 5-1 at Hearts, with Billy King, Lewis Horner and Alex Fisher coming in for Jake Mulraney, David Raven and Scott Boden.
St Johnstone made two switches, Chris Kane and Steven Anderson replacing Murray Davidson and Dave Mackay in the starting line-up.
Inverness had the early opportunities in a quiet start, with Draper planting a header goalwards that Alan Mannus did well to parry, while Blair Alston blocked a Billy King shot.
Draper was proving a menace to the Saints defence, regularly finding pockets of space and creating chances for team-mates. He anchored a move that saw Liam Polworth fire wide from 25 yards and teed up Fisher, whose looping header was tipped over by Mannus.
As the first half wore on Inverness grew more dominant, with King stinging Mannus' palms with a low drive and Fisher scuffing a Draper centre straight into the goalkeeper's grateful arms.
Clever wide play by Polworth deserved better than an uncontested ball flashing across the face of goal and Inverness would undoubtedly have felt they should have been ahead at the break.
The closest St Johnstone came was a Chris Kane effort that was blazed high over the top.
Draper finally gave Inverness the lead they deserved on the stroke of the hour, rising highest to glance home a Greg Tansey free-kick.
It was just reward for Draper, who had been the stand-out player throughout the game - and he celebrated his goal with gusto.
Substitute Davidson guided a header wide as the visitors sought to find a way back into the game - and they got their goal with 11 minutes to go.
Foster, who played Inverness just a fortnight ago with Ross County, was fed the ball by Steven MacLean and from 20 yards, he arrowed his shot straight into the top corner.
After late pressure, where Draper had another header saved by Mannus, Meekings pounced to cap a memorable victory.
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