Tunisia threatened on the counter attack but Boumnijel almost gifted Russia a third goal, Dmitri Sychev shooting wide after the keeper had dropped the ball.
The win takes Russia to the top of the group following yesterday's 2-2 draw between Japan and Belgium.
Russia coach Oleg Romantsev said: "It was a good game for the fans but there will be reprimands for my players.
"We cannot allow our opponents to create so many chances and we created chances ourselves but did not finish them off.
"So we have work to do in both penalty areas."
Tunisia joint coach Ammr Souayah said conceding two quick goals had given his side too much of a mountain to climb.
"It's true that we held out well for an hour but we conceded two quick goals against a very strong and experienced Russian team and it was difficult for us to come back.
"Many neutral observers consider us the weakest team in the group, but we have to recover our spirits and prepare for the next game (against Belgium). It's going to be the determining game for us."
