The Tigers doubled their goals-for tally in home league matches this season and wrapped up their third win in a row against a Pompey side who saw Luke Varney sent off.
Koren, celebrating the birth of his daughter earlier in the week, scored with a stunning long-range effort after just 11 minutes and the hosts held on for a deserved victory.
Tom Cairney's early long-range effort dipped just over the bar as Nigel Pearson's men made a strong start.
Koren then cracked the post with a shot from distance and although Martyn Waghorn was on hand to tap in the rebound, his celebrations on his home debut were cut short by an offside flag.
But moments later Koren fired an unstoppable effort beyond the reach of Jamie Ashdown from 25 yards.
The visitors tried to respond through the industrious Erik Huseklepp, who earned a corner off Joe Dudgeon after good work in the box, but the set-piece came to nothing.
Koren had to get his hands dirty in his own box as the Slovenian was forced to clear after Jack Hobbs' header had run loose.
The Tigers continued to threaten and the instrumental Koren fed Cairney in the 37th minute but the midfielder's attempted chip was gathered by Ashdown.
Adriano Basso had a lucky escape just before the break after appearing to misjudge Benjani's looping header, only to be spared embarrassment as David Norris was ruled to have impeded his hapless efforts to claim the ball.
A quick break by City required some smart defending by Greg Halford as he blocked Robbie Brady's menacing effort behind for a corner just before half-time.
Waghorn, on loan from Leicester, dragged a shot wide early in the second half when he had Aaron Mclean waiting for the square pass.
With Steve Cotterill seeking to end his side's three-game winless run, he introduced the experienced Kanu from the bench in an attempt to give Pompey a much-needed boost.
The striker took advantage of a deep-lying City defence to jink inside and feed Hayden Mullins, whose attempt to force his way through the home defence was thwarted by a strong intervention from Hobbs.
But Pompey's hopes of turning things around suffered a major blow in the 78th minute when Varney was, perhaps harshly, dismissed for a foul on James Chester, the referee suggesting his studs were showing.
The home side demonstrated a calmness in possession which allowed them to comfortably see the game out and add three points to their steadily-improving tally.