Griezmann endured a disappointing debut in last weekend's surprise 1-0 defeat to Athletic Bilbao, but the €120million signing from Atletico Madrid – leading the line in the absence of Lionel Messi and Luis Suarez – announced himself to the Camp Nou faithful in style.
Nabil Fekir had given Betis a shock 15th-minute lead before Griezmann turned the game on its head either side of half-time with two fine finishes.
Barcelona did not look back after his second, adding further goals through Carles Perez – his first for the club – Jordi Alba and Arturo Vidal before Loren Moron's wonderful late consolation for the visitors.
Barca started sharply, going close through Sergi Roberto and Griezmann, but it was the visitors who struck first.
The host lost possession cheaply in the middle of the pitch, allowing Loren Moron to feed the marauding Fekir. The former Lyon man pulled away from Alba and dispatched a clinical left-footed strike from 16 yards into Marc-Andre ter Stegen's bottom-left corner.
Barca came agonisingly close to restoring parity 10 minutes before the interval when Rafinha's close-range effort was diverted narrowly past the post by the outstretched leg of Sidnei.
It did pull level in the 41st minute, however, Griezmann stretching superbly to meet Roberto's clipped ball over the top and volley past Dani Martin.
Roberto was the provider again for Griezmann's second in the 50th minute, the France international whipping a glorious left-footed strike past Martin from 18 yards.
Barcelona was well and truly in the mood and scored a third six minutes later when Perez skipped past his marker and coolly slotted past a helpless Martin from just inside the area.
The Betis goalkeeper picked the ball out his net again just four minutes later, this time from Alba. The left-back surged into the penalty area, collected Sergio Busquets' pass and slid past the beleaguered Martin.
Barca eased off in the final 30 minutes, but they did add a fifth in the 77th minute when substitute Vidal expertly steered into the roof of the net from 10 yards after being picked out by Griezmann.
The final goal was one from Betis, with Moron firing a wonderful strike into the top corner from 25 yards in the 79th minute, although it did little to detract from a fine Barca win.