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.
Ipswich recalled striker Michael Chopra and midfielder Guirane N'Daw as they looked to extend their unbeaten run to six matches.
Hull defender Jack Hobbs made a surprise return to the side after an ankle injury threatened to cut short his season, while David Meyler returned to midfield and Simpson stepped in for injured striker Gedo.
The visitors, chasing the win that would keep them at least three points ahead of third-placed Watford for the second automatic promotion slot, dominated the early exchanges.
James Chester lashed an early shot over the crossbar following a corner and Simpson's goalbound effort was blocked by Ipswich defender Aaron Cresswell.
Ipswich had been restricted to one mis-hit shot from Chopra before Hull took a 28th-minute lead through Brady's penalty.
Referee Oliver Langford pointed to the spot following N'Daw's challenge on Meyler and Brady coolly side-footed the ball, left-footed, into the bottom corner.
Hull went close to doubling their lead on three occasions towards the end of the first half after Ipswich midfielder Jay Tabb was forced off through injury and replaced by Wordsworth.
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 substitute Wordsworth, who steered his header into the bottom corner from six yards.
Home striker Daryl Murphy then headed Wordsworth's free-kick over the crossbar, while Carlos Edwards headed wide as Ipswich's confidence soared.
Hull responded through Brady's shot from outside the area that flew wide, but the visitors were dealt a blow when midfielder Corry Evans went off injured, with Ahmed Fathi stepping off the substitutes' bench.
Visiting goalkeeper David Stockdale saved brilliantly to keep out 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.
Hull striker Matty Fryatt made his first league appearance of the season in the closing stages following an Achilles injury as the visitors held on to their lead.
- Sports & Recreation
- Robbie Brady
- Anthony Wordsworth