Wissam Ben Yedder was on top form as he netted a hat-trick against the Slovenian champions, with the La Liga side in complete control throughout at the Ramon Sanchez Pizjuan.
And Berizzo, who had to watch on from the stands after he was sent off in the matchday one draw with Liverpool at Anfield, was enthused by his side's display.
"It was a convincing win, we managed the ball very well," Berizzo told reporters.
"We won the game with a lot of authority, when we increased the speed, we became unstoppable.
"We accelerated at the right moments, and that's where the goals were achieved, when we accelerated and changed the pace of the attack."
Ben Yedder opened the scoring 27 minutes in following superb play from Joaquin Correa, before doubling his tally prior to the interval.
Sevilla were frustrated in their efforts for a third until late in the second half, when Ben Yedder fired home from the penalty spot moments after Jesus Navas had struck the crossbar.
Maribor toiled throughout, with captain Marcos Tavares occasionally threatening on the counter, but coach Darko Milanic says the team can take heart from their display, though he wanted his side to be more proactive on the ball.
"This was a quality opponent. In the first half we doubled-up on certain players and that is hard to sustain in the long run," Milanic said.
"We were a bit too shy when we had the ball. They didn't surprise us, we knew they would be a tough nut to crack.
"We wanted to close down their midfield and as soon as our concentration dropped, we conceded."