Augbsurg lost ground on their rivals for a Champions League spot as Werder Bremen claimed a 3-2 victory for a fifth successive Bundesliga win.
However, Klavan gifted Werder the chance to retake the lead two minutes later when he fouled Davie Selke in the box, and Franco Di Santo capitalised to score his 10th of the season from the penalty spot.
Augsburg continued to look threatening, but their task was made even more difficult before the break when another precise set-piece delivery from Junuzovic allowed Theodor Gebre Selassie to head home his side's third of the afternoon.
Junuzovic twice went close to adding a goal to his impressive display, but his first effort crashed against the crossbar, before a powerful strike on the hour mark drifted just over.
Augsburg set up a dramatic finish by reducing the deficit 11 minutes from time thanks to Tobias Werner, who latched onto a loose ball in the box to fire home after Werder had failed to clear a corner.
However, the hosts looked comfortable in the closing stages and Junuzovic struck the woodwork again from a free kick as they went close to extending their margin of victory under new manager Viktor Skrypnyk.