Kurz's mood soured further when Spanish midfielder Mandi scored the first Phoenix goal in four matches, volleying a bounced shot from the top of the box in the 14th minute.
The hosts were unfortunate to concede on the stroke of half-time when defender Tom Doyle headed the ball into his own net from close range.
"In the first 30 minutes it wasn't good enough in my opinion," Kurz said.
"The equaliser before half-time was very important; it was good for us.
"After half-time we played more comfortable, more ball speed and created a couple of good chances. In the second half we were the better team."
Ilso scored his first two goals for the club, meeting Michael Marrone's curled delivery with a diving header in the 47th minute before latching on to Kitto's cross in the 64th.
The latter initially appeared to be a second own goal from Doyle but the Dane was favoured by a VAR ruling.
The Reds weren't always comfortable but had more attacking class than the defence-minded Phoenix, bouncing back from a loss to Perth before the international break to record their second win.
It is a third straight loss for the Phoenix, who lacked creativity without young All Whites playmaker Sarpreet Singh.
They are without a win since their opening defeat of Newcastle and have conceded eight goals in their last three games.
Coach Mark Rudan rated it one of the better performances from his team but lamented an ability to cope with crosses, the source of all three Adelaide goals.
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"We were disciplined and organised and deserved to go 1-0 up, but the game was won and lost just before and just after half-time," he said.
"Individual mistakes and errors have cost us tonight."
Rudan handed first starts for the season to forward Nathan Burns, defender Ryan Lowry and Polish midfielder Michal Kopczynski, who played most of the game in an unfamiliar centre back role.
Taylor Regan replaced Michael Jakobsen, who has muscle soreness but the Denmark centre back is likely to be fit to face Brisbane Roar next week.
AAP