The 20-year-old Slovakia international arrives in a deal worth around €12.5m, becoming the fifth most expensive signing in the club's history behind Deniz Undav, Ermedin Demirovic, Bilal El Khannouss and Badredine Bouanani.
VfB Stuttgart beat competition from Bundesliga rivals Hoffenheim to land Sauer, and are also working to finalise a move for 21-year-old forward Pejcinovic, with Wolfsburg holding out for a fee closer to €25m against Stuttgart's most recent offer of around €22m including bonuses.
Both incoming deals are effectively being financed in advance of departures still to come, with head coach Sebastian Hoeness needing to balance the books after Stuttgart again secured Champions League qualification last season.
Winger Bilal El Khannouss, only made a permanent signing from Leicester City for around €18m earlier this year after an impressive loan spell, has emerged as a leading sale candidate given Stuttgart's need to generate transfer income, with his stock having risen further after a strong World Cup as a regular starter for Morocco.
Right-back Leonidas Stergiou is also free to leave after two years at the club, with former side St Gallen showing interest even though no agreement has yet been reached.
Elsewhere, several fringe players have been told they can leave or are actively seeking exits themselves, including Mirza Catovic, Laurin Ulrich and Christopher Olivier, while a buyer is also being sought for Jovan Milosevic following his loan spell at Werder Bremen last season.
Jeff Chabot is wanted by Besiktas, with Stuttgart said to want at least €20m for the centre-back, while forward Ermedin Demirovic has drawn interest from Turkey, Leeds United and Juventus, though he could yet be squeezed out of the squad regardless of any bid if Pejcinovic's arrival goes through.
El Khannouss represents the biggest potential windfall of the lot, with Stuttgart understood to be open to discussions from around €60-65m for the Morocco international.