Visitors Barca, who had lost their prior two league matches, fell behind in the first half after a goalkeeping blunder by Neto allowed Luis Rioja to open the scoring.
Alaves had Jota sent off for a second bookable offence just after the hour mark, however, and the Blaugrana quickly equalised with a fine finish from France striker Griezmann.
But Ronald Koeman's side could not forge a decisive second despite late pressure, instead seeing their winless LaLiga run extended to four matches as they languish down in 12th.
FULL TIME pic.twitter.com/WHJLEQGWzZ— FC Barcelona (@FCBarcelona) October 31, 2020
Barca had begun brightly and Ansu Fati raced clear of the Alaves defence early on but could only strike his effort wide.
However, the visiting back line looked vulnerable on the counter and Rioja brought a smart save out of Neto after fine play by Deyverson.
It was from another Alaves breakaway that Barca found themselves behind after 31 minutes, although the goal came courtesy of self-inflicted Blaugrana errors rather than brilliance from the home side.
Gerard Pique made a weak, dangerous pass to Neto rather than clearing and, under pressure, the Barca goalkeeper got himself in a tangle which allowed Rioja to steal the ball and tap into the vacant net.
Koeman made a triple-change at the interval, but his team were not level until after Alaves were reduced to 10 men following Jota's high boot on Pique.
Griezmann took full advantage just over a minute later and stylishly clipped the ball over Fernando Pacheco as the Alaves goalkeeper raced off his line.
Pacheco saved superbly from Lionel Messi, but Griezmann soon thought he had the vital second, only for Trincao to be flagged offside in the build-up.
And Alaves were indebted to Pacheco as they hung on through a tense finale, their keeper saving from Miralem Pjanic and Sergino Dest to preserve a point.
What does it mean? More scrutiny on Koeman
The Catalan giants managed to avoid that unwanted statistic with a draw here, yet their new coach will continue to find himself under the microscope as his side already trail leaders Madrid by eight points - despite the clubs' contrasting European fortunes.
The look of despair on the Brazilian goalkeeper's face after his dismal error spoke volumes. Long periods of inactivity are common for a Barca keeper but a lapse in concentration proved costly here.
Neto made just two saves in the match and looked shaky throughout. The return of first-choice goalkeeper Marc-Andre ter Stegen, who recently resumed training following knee surgery in August, cannot come soon enough for Koeman.
In many regards, this match was a tale of two goalkeepers, with Neto's opposite number quite brilliant between the posts for Alaves.
Pacheco commanded his box with two clearances, vociferously organised his defence and contributed seven crucial saves. There was little more he could have done for Griezmann's goal.