Forward Mahrez, who plays for Al Ahli in Saudi Arabia after moving from Manchester City, struck twice in Rabat, Morocco before Ibrahim Maza netted his first international goal.
The 34-year-old Mahrez, now Algeria's all-time top scorer at the Cup of Nations with eight goals, has faced scrutiny after the Desert Warriors, champions in 2019, exited at the group stage in the last two editions.
"This wasn't about answering criticism," Mahrez told reporters. "I'm used to it.
"My reply is to concentrate on my game on the pitch. We came in with a clear plan to play without overthinking, to win this one and the rest of our group matches to make the path easier."
Algeria missed out on the last two World Cups but have qualified for the expanded 2026 tournament in the U.S., Mexico and Canada.
Mahrez won the rebranded AFC Champions League Elite with Al Ahli this year, when the Jeddah club beat Japan's Kawasaki Frontale 2-0 to lift their first continental title.
"It's good to score - it brings confidence - but the team has to keep lifting our level so we're more consistent," Mahrez said.
"The focus turns immediately to Burkina Faso. We need to raise our game and be more stable from the next match."
Algeria lead Group E with three points and meet Burkina Faso on Sunday.
Earlier on Wednesday, Burkina Faso scored twice deep into second-half stoppage time to beat 10-man Equatorial Guinea 2-1.