Nabil Fekir had sent Betis into a shock 15th-minute lead on the night - at which point Barca were staring at a second successive defeat having lost their 2019-20 opener at Athletic Bilbao last weekend.
Griezmann then struck wide of the post from distance with Barca making most of the early running against Betis, who lost their league opener at home to Real Valladolid last weekend.
Another half-chance came for Sergi Roberto when the midfielder met a cross from Alba, only to get his contact all wrong and head high rather than towards the goal.
Betis beat Barca 4-3 at Camp Nou last season, though, and the Seville outfit looked confident and comfortable early on despite having to concede a lot of the possession.
The visitors took the lead from their first breakaway in the 15th minute when Loren played a clever pass into Fekir, who picked out the bottom corner with a brilliant finish from a tough angle.
Ernesto Valverde's side had a lot of the ball in the period that followed - Betis were far from unnerved, though, and did not have to work too hard to keep their one-goal lead as the first half developed.
Gerard Pique was perhaps a touch fortunate to only pick up a yellow card in the 32nd minute when he brought Loren to the ground with a last-man challenge outside the Barca box.
The resulting set piece from Fekir had to be saved by Marc-Andre ter Stegen, with the fast counter another indication of the threat that the visitors could provide.
Rafinha had a golden chance to equalise late in the first period after being found by Alba inside the Betis box, but the Brazil international saw his effort deflected wide of the post by Sidnei.
Barca did level the scores in the 42nd minute, though, when Griezmann poked a brilliant pass from Roberto past the Betis goalkeeper to register his first competitive goal for the club.
Perez had the home side's first chance of the second period after breaking into a dangerous position, but the 21-year-old forward fired wide of the post.
Valverde's side did take the lead shortly after, however, when Griezmann picked out the bottom corner of the net with an excellent curling strike from distance. Barca desperately needed someone to step forward in the absence of some key players, and the summer arrival certainly answered the call.
Perez then sent a powerful effort just wide of the Betis post shortly after as the home side looked for a third.
The forward was not to be denied, though, and made it 3-1 Barca in the 56th minute when he picked out the bottom corner with a clever effort after working space just inside the Betis penalty box.
The champions were not finished, however, and scored a fourth shortly after when Alba collected a clever pass from Sergio Busquets before squeezing the ball past the away side's stopper.
Betis had impressed for long spells of the first period but were unable to deal with the rise in tempo as Valverde's side flexed their muscles in front of their own supporters.
The home side's fifth of the match arrived in the 77th minute when Vidal found the top corner with a brilliant strike off a Griezmann assist, before 16-year-old Anssumane Fati entered the field for his debut.
The goal fest was not finished, though, and Loren netted an absolute stunner for Betis in the 79th minute - finding the top corner from distance to finish the scoring at Camp Nou.
Fati almost made it a dream night late on when he looked to pick out the bottom corner from a narrow angle, only to watch the ball move just wide of the post.
Next up for Barca is a trip to Osasuna on Saturday, while Betis will be looking for their first points of the season when they host Leganes.
BARCELONA (4-3-3): Ter Stegen; Semedo, Pique, Lenglet, Alba; De Jong, Busquets (Vidal 74'), Roberto; Perez (Fati 78'), Griezmann, Rafinha (Junior 81')
REAL BETIS (4-2-3-1): Martin; Emerson, Bartra, Sidnei, Alfonso; Carvalho, Guardado (Kaptoum 61'); Fekir, Canales (Lainez 75'), Tello (Joaquin 61'); Lorenbody check tags ::