The Terrors finished third in the Scottish Championship and had already seen off Greenock Morton and Falkirk in two-legged ties ahead of their encounter with the Accies.
After a goalless first leg at Tannadice, a solitary goal from Greg Docherty was enough to settle the score in Hamilton's favour as McKinnon's men were denied an immediate bounce back to the top flight.
"The six games has taken its toll, no doubt about it," he told BBC Sport afterwards. "It was an evenly-contested game but you could see the legs were gone in the second half. We didn't have the same intensity to our pressing.
"We were running on empty and unfortunately we've just come up short at the final hurdle.
"We have to work hard over the summer to attract better quality than we've got. We've started the recruitment process well with James Keatings and Billy King. If we can maintain that standard, we'll give ourselves a great chance next season."
Victory for United's opponents today guaranteed them an unprecedented fourth consecutive season in the Premiership.