The Northern Ireland international drilled beyond Allan McGregor with just two minutes remaining to ensure it was honours even.
It meant Hull were denied a third straight win, though they did at least return to fourth place after being leapfrogged earlier in the day by Derby County.
With a home game against Brentford to come on Tuesday, City remain favourites to guarantee home advantage in the second leg of the play-offs.
Nevertheless, this draw felt like a missed opportunity after two goals in first half stoppage time from Abel Hernandez and Tom Huddlestone had seemed to put the home side on their way towards all three points.
Leeds, though, will also have left the KC Stadium wondering what might have been? Not only was it two big lapses in concentration at the end of the first half that proved their undoing but the visitors also had a penalty saved.
Chris Wood, United’s scorer on 14 minutes, was the guilty party, though much credit does have to go to Allan McGregor for a quite exceptional save low to his left.
Dallas’ late equaliser stretched Leeds’ unbeaten run to four games and kept alive hopes of a possible top ten finish.
United’s opener had been begun and finished by Wood, who after spreading play to Charlie Taylor on the left flank dashed into the penalty area to meet the cross with a bullet header.
The Leeds top scorer was helped by being totally ignored by the Hull defence, who were drawn towards the crosser to leave the centre of the six yard box totally unmarked.
It came on 15 minutes and followed a decent opening from the Tigers, who went close through efforts from Abel Hernandez and Tom Huddlestone either side of a Chuba Akpom shot that flew straight into the hands of Marco Silvestri.
After Wood had notched his 12th goal of the season, Curtis davies headed just over for the hosts from a Shaun Maloney corner and then Harry Maguire drilled a long distance effort wide.
Neither unduly worried Silvestri but the Italian did have one anxious moment when a Moses Odubajo shot looped off Taylor and on to the top of the net.
That appeared to be that only for two goals in stoppage time to transform the contest.
First, Abel Hernandez tapped in from a Robert Snodgrass cross and then, with Leeds still reeling, Tom Huddlestone’s drilled shot found the corner of the net.
It was an incredible turnaround but one that Moses Odubajo threatened to undo eight minutes into the second half with a rash sliding tackle on Taylor that left referee Iain Williamson pointing to the spot.
Allan McGregor, however, spared his team-mate’s blushes with a fine save low to his left to deny Wood.
Hull had a couple of chances to put the game to bed but Hernandez’s shot was blocked and Snodgrass fired over.
Wood headed straight at McGregor with a late chance before Dallas rescued a point with a pinpoint low drive into the bottom corner of the net following a neat pass from Jordan Botaka.
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