With the game in the balance at 2-1, Thiam settled the issue with a blockbuster of a strike ten minutes from time with his glorious effort flying past James Shea.
It was an effort worthy of winning any contest, with the Reds' victory continuing their upward march and also extending their unbeaten league home sequence to ten matches, with the run stretching back to March 17.
A second goal of the game from Luton forward James Collins on 85 minutes set up an anxious finale, but the hosts prevailed - watched on by former manager Paul Heckingbottom, who was on media duties for Sky Sports.
A clinical early strike from Potts - his fifth of the campaign and third in the space of a week - and a noteworthy goal against his former club from McGeehan, his first goal in almost exactly a calendar year - provided the hosts with a dream start in the first half-hour.
Collins pulled one back from the spot in first-half stoppage time after Ben Willams rashly stuck out a leg and felled Harry Cornick.
McGeehan got the nod to step into the fray for Kenny Dougall and did not disappoint in an impressive first half in which the Reds produced some exquisite moves in their new-look diamond - after matching up with the visitors.
It yielded two super goals and some sublime transitional football along the way.
On the debit side, the hosts looked susceptible down their left and paid the price just before the interval when the Hatters pulled one back.
The game was just five minutes old when Potts crowned a productive seven days by firing a crisp opener - following on from his two well-taken strikes against Peterborough.
After being supplied by Jacob Brown, starting up front with Kieffer Moore sidelined, Potts needed no second invitation to surge forward and firing a fine low drive past James Shea.
It was the perfect start for the Reds, who went close to a second when Dimitri Cavare's drive deflected just wide before Luton offered a scare when James Justin's cross-shot flashed across goal.
An offside flag - with Luton insistent their player was level - then saved the Reds after Collins netted a rebound after Jack Walton spilled a strike from Pelly-Ruddock Mpanzu.
It proved key with McGeehan soon calmly steering home the hosts' second on 27 minutes after fine work by Mike Bahre and George Moncur, enjoying a fine half.
Sonny Bradley almost pulled one back with a header following Jorge Grant's free0kick before Grant was inches away with a expertly-taken free-kick.
Luton were then afforded their lifeline from the spot to set up an intriguing second period.
It was Barnsley who forged the first meaningful chance with McGeehan seeing his close-range header clip the bar following a wind-assisted inswinging free-kick on the right from Alex Mowatt.
Luton soon switched systems to a 4-2-3-1, with the Reds replying in kind in a technically accomplished and watchable encounter high on quality.
The Reds pushed for a killer third with Potts' low drive gathered by Shea after a Mowatt corner was not cleared.
Heading into the final quarter, the game was on a knife-edge, with Thiam providing some spectacular respite.
But it proved only temporary when Collins fired home a rebound after substitute Kazenga LuaLua's left-wing cross struck the woodwork.
Barnsley: Walton; Cavare, Pinnock, Lindsay, Williams (Pinillos 73); McGeehan, B Potts, Mowatt, Moncur (Hedges 83), Bahre (Thiam 68); Brown. Substitutes unused: Kendrick, Bird, Jackson, Adeboyejo.
Luton Town: Shea; Stacey, Pearson, Bradley, D Potts; Rea (Hylton 79), Mpanzu, Justin, Grant (McCormack 58); Cornick (LuaLua 72), Collins. Substitutes unused: Shinnie, Jarvis, Sheehan.