Sascha Molders gave the visitors the lead in the 10th minute, but as soon as they lost Ragnar Klavan to a red card six minutes later there was only going to be one winner at the Veltins Arena.
Boateng scored the penalty which followed Klavan's dismissal before Hungarian forward Szalai struck twice, breaking Augsburg's resistance.
Young Max Meyer completed the rout with his first Bundesliga goal four minutes from time as Schalke leapt above their opponents in the standings.
Molders ended his 10-game goal drought to put a bright and confident Augsburg in front, turning in a deflected left-wing cross from close range.
Schalke were gifted a route back into the game when Klavan hauled Szalai down inside the penalty area five minutes later, earning a red card before Boateng sent his spot-kick beyond the reach of Alexander Manninger.
Augsburg coach Markus Weinzierl was forced to alter things, taking forward Halil Altintop - a Gelsenkirchen native - out of the action and giving a debut to Hong Jeong-ho.
It nevertheless only seemed a matter of time before Schalke struck again with Szalai obliging from Japanese full-back Atsuto Uchida's pass.
Felipe Santana hit the post before Szalai scored his second of the game, breaking Augsburg's resistance with a far-post lunge to convert Dennis Aogo's cross.
Meyer, who turned 18 just a fortnight ago, put the cherry on top for the Royal Blues with a composed finish on an altogether positive afternoon for Schalke.