They signed Liverpool midfielder Bobby Clarke earlier in the window for a fee believed to be worth about £10m.
Red Bull Salzburg are top of the Austrian Bundesliga with three wins from three and qualified for the Champions League group phase this week.
“I would like to win the league and impress in the Champions League too with my new team,” Bajcetic said.
“An important reason for my move is that Salzburg have focused on developing young players for many years – it’s a club with lots of potential.
“Pep Lijnders, who I know from Liverpool and value, is the perfect coach for that. I am looking forward to it all.”