Sweden badly needed a goal and five minutes after half-time they deservedly got it, defender Augustinsson popping up from left-back to volley in.
Originally called out for offside, the goal was given after video review.
Sweden, who have rallied behind Durmaz, can qualify with a victory or a draw against Mexico if Germany lose to South Korea. Sweden will play in the knockout stage for the first time in a dozen years despite blowing a 1-0 lead on 10-man Germany in the previous match, the 2-1 defeat coming on virtually the final kick in injury time.
But they needn't have anxious, South Korea provided an equally stunning 2-0 win in Kazan to send the holders Germany home.
"When we heard after the match that this had occurred, the squad was annoyed", Granqvist revealed. Granqvist stepped forward, and the Sweden captain found the top corner with a wonderful effort to double his team's lead. Javier said no one inside Mexico's locker room is disappointed about not having already earned its place in what would be its seventh consecutive spot in the round of 16.
Sweden wins Group F and will face Switzerland in St. Petersburg on Tuesday.
Mexico's fans in Russia can go ahead and book their accommodation in the southwestern Russian city of Samara, where Mexico will take on its round of 16 opponent on July 2, while those back home will no doubt be once again glued to their screens. Hernandez notched his 50th worldwide goal in Mexico's 2-1 win over South Korea on Saturday.
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After the match, a delighted Andersson said: "We worked as a team. However and nevertheless, I am very hurt".
"We want the players to recover mentally and physically (from this match)".
At the interval, with the game also goalless in Kazan, Mexico and Germany were going through.
A confident Mexico team played Sweden in the final game of the group stage.
Sweden were on course for the last 16 until Toni Kroos stepped up with a wonderful victor in the fifth minute of injury time to rescue the reigning champions 2-1 in a Saturday-night thriller.
Even in Yekaterinburg, almost 500 miles away, the Mexican fans expected Germany to score. "We haven't forgotten hearing the boos at the Azteca, and that has strengthened us mentally", Ochoa said.
Mexico attacked for the rest of the first half, with Lozano's shot blocked, Vela blasting over the bar and Miguel Layun's soft shot easily gathered.