Paraguay initially took the lead on a fourth-minute Oscar Cardozo finish from the penalty spot before Derlis Gonzalez doubled that advantage in the 56th minute.
The wheels fell off from there, though, with a 68th-minute Almoez Ali goal and an own goal nine minutes later from Juan Rodrigo Rojas sealing a split of the points.
Reflecting on Paraguay's Group B draw in Rio de Janeiro, head coach Berizzo told reporters: "I think it would make sense to play a Copa America with teams from all the Americas.
"Are you asking me about invited nations, right? About Qatar and Japan? I think we should imagine a whole Copa America, with Central America and North America playing all the same tournament.
"I never saw Europe inviting any South American team to compete.
"Don't take it wrong, maybe it sounds a little acidic today, just after playing against [Qatar]. But I firmly believe Copa America should be played by all the American teams in one tournament – CONCACAF and CONMEBOL together."
#VibraOContinente com o resultado entre. A Albirroja e os Catarianos empataram no fim da 1ª rodada do Grupo B na @CONMEBOL #CopaAmerica . Mais estatísticas aqui: https://t.co/mYYF6r9PXt pic.twitter.com/0WlytxfSgN— Copa América (@CopaAmerica) June 16, 2019
Paraguay looked to be on track for a winning start thanks to Oscar Cardozo's fourth-minute penalty before Derlis Gonzalez added an unstoppable second.
Qatar, however, fought back in the second half via Almoez Ali and Juan Rodrigo Rojas – who headed a Boualem Khoukhi effort into his own goal.
"This result disappoints us all, we all want to win, from the fan to the coach," Berizzo said. "We need to mind our rival, our reality, a lot of things that explains a lot about this situation.
"This is a new cycle for the team, there is no way I didn't know Copa America was coming to this date, so I cannot make any excuse of it if we cannot play well, when we play well I will tell you we did it.
"Today we had a hard time finding the ball, especially in the middle of the field, and especially after being 2-0 up. If we would have this done, we should get another result."