All the goals arrived in an explosive first half and it was the visitor which drew first blood, Bozanic opening the scoring after eight minutes.
Aarau, which lost 3-1 at champion Basel in its first game last weekend, responded superbly.
Olivier Jackle equalised in the 13th minute and further goals from Remo Staubli, Davide Calla and Artur Ionita had turned the game in its favour with eight minutes of the first half still on the clock.
Bozanic grabbed his second goal five minutes before the interval to give Lucerne hope heading into the second period, but it had no further joy after the break and ended the game with 10 men after the last-minute dismissal of Dimitar Rangelov.
Also in the Swiss top flight, Young Boys won 3-1 at Lausanne Sports to move top of the fledgling standings.
Gonzalo Zarate's first-minute goal set the tone and Raphael Nuzzolo and Scott Sutter had stretched that advantage to 3-0 by the 40-minute mark.
Half-time substitute Yannis Tafer made an immediate impact by netting in the first minute of the second period to give Lausanne a glimmer of hope, but that was as good as it got.