The Serbian winger, 25, joined the Saints on a four-year deal from FC Twente on Tuesday afternoon.
"It was very important to get this signing done – not only for me, but for the rest of the squad and for our fans. It's fantastic news," Koeman told Southampton's official website.
"We've lost some good players but we want to continue the ambition and philosophy of the club, so it's a great signal to have brought in our first player.
"He's a great player. I know him very well from Holland, where he had a great season last year. We now look forward to bringing in more players because we want to continue and maybe even better what Southampton did last season."
Tadic joins Southampton for an undisclosed fee.