The Oakwell outfit, who had lost their previous five matches at the start of a season, began the campaign on a emphatic winning note, with Thiam's double providing the icing following a dominant first-half performance.
Thiam picked an opportune moment to grab his first goals on home soil, with the Sengelese netting with identkit strikes after 20 and 29 minutes.
Thiam, whose only other Reds goal arrived last September, netted his opener after unleashing a fierce low angled strike which flew past ex-Reds target Simon Eastwood after good approach work from Dimitri Cavare.
The forward then repeated the trick after neat play by Tom Bradshaw and Potts.
The strikes were just reward for an outstanding half from Barnsley, which was brimful of energy, intensity and desire.
The Reds' efforts were lapped up by the home faithful, who broke into a chorus of 'Daniel, give us a wave' with ten minutes of the half to go.
Thiam almost completed his hat-trick just before the break, only for Eastwood to race out and gather with Oxford's best chance coming when Jon Obika's low shot was gathered by Adam Davies.
The Reds failed to hit their first-half heights on the restart as Oxford came into the game, but the hosts still had their moments, most notably when debutant Kenny Dougall's audacious long-range effort from the half-way line was turned away in the nick of time by Eastwood.
The hosts then sealed proceedings with an emphatic finish into the roof of the net from Potts 13 minutes from time, which came shortly after the U's went close when Davies parried sub Rob Hall's fine long-range strike.
Barnsley: Davies; Cavare, Pinnock, Lindsay, Pinillos; Potts, Dougall, Mowatt (McGeehan 78), Thiam (Moncur 70); Moore (Adeboyejo 82), Bradshaw. Substitutes unused: Walton, Jackson, Isgrove, Bird.
Oxford United: Eastwood; McMahon, Dickie, Nelson, Garbutt; Carruthers (Smith 21), Brannagan, Ruffels, Henry (Hall 63); Mackie (Whyte 82), Obika. Substitutes unused: Stevens, Norman, Mousinho, Raglan.
Referee: R Joyce (Cleveland).
Attendance: 12,820 (1,279 Oxford supporters).