Koren, a second-half replacement for Jay Simpson, headed home Robbie Brady's cross in the 83rd minute after Ipswich substitute Anthony Wordsworth had cancelled out Brady's first-half penalty.
Victory for the Tigers lifts them six points clear of nearest challengers Watford, who slipped to a surprise defeat at Peterborough, with four games left to play.
Ipswich had lifted themselves clear of immediate relegation trouble with just one defeat in their previous eight matches, but they were overrun in the first half by Steve Bruce's side, who secured their 11th away win of the season.
Town had been restricted to one mis-hit shot from Michael Chopra before Hull took a 28th-minute lead through Brady's penalty.
Referee Oliver Langford pointed to the spot following Guirane N'Daw's challenge on David Meyler and Brady coolly side-footed the ball, left-footed, into the bottom corner.
Defender Abdoulaye Faye headed a corner straight at Ipswich goalkeeper Scott Loach, who then pulled off a fine save to keep out Brady's shot from outside the area and did well to hold George Boyd's angled effort from 12 yards.
Ipswich regrouped during the interval and after gaining an immediate foothold after the restart the home side struck a 55th-minute equaliser with their first attack on goal.
Defender Luke Chambers broke forward down the right and his cross was met by Wordsworth, who steered his header into the bottom corner from six yards.
Visiting goalkeeper David Stockdale saved brilliantly to keep out Daryl Murphy's thumping shot from inside the area and just as it appeared Ipswich had done enough to earn a point, Koren ghosted in to head home Brady's cross with seven minutes left.
- Sports & Recreation
- Robbie Brady
- Anthony Wordsworth