Daly slotted home almost two minutes after regulation time had elapsed following a clever pass from David Goodwillie, who had levelled just before the interval.
Flavio Paixao had claimed a richly-deserved opener for a Hamilton side featuring just one natural defender and the bottom club were denied a strong penalty claim in the latter stages of the second half.
The Tayside club are now just two points behind fourth-placed Kilmarnock following wins over Inverness, Aberdeen and luckless Accies, who remain seven points adrift and without a win since October.
United manager Peter Houston admitted he would leave players out with Sunday's Scottish Cup quarter-final against Motherwell in mind, although he had no choice but to lose Craig Conway after the winger aggravated a thigh injury in Monday's 3-1 win over Aberdeen.
Danny Swanson and Prince Buaben dropped to the bench with Johnny Russell, Andis Shala and David Robertson promoted.
Mark McLaughlin joined Martin Canning on the injured list for Hamilton to leave them short of defenders.
Their central defence comprised David Elebert and Simon Mensing, who made his name as a midfielder, and wingers Dougie Imrie and Jim McAlister started at full-back.
Hamilton's makeshift defence was rarely tested in the opening half hour as they controlled possession with Paixao, Gary McDonald and Ali Crawford all supporting centre-forward Mickael Antoine-Curier.
Accies created the first chance when Paixao headed Imrie's cross into the path of McDonald, who was closed down as he aimed a side-footed effort towards the bottom corner from 15 yards, with Dusan Pernis getting down to hold.
The visitors came closer when Barry Douglas cleared Mensing's header off the line following Crawford's corner.
Hamilton soon took the lead from another corner in the 26th minute. Imrie swung the ball beyond the far post and Mensing headed over Pernis into the goalmouth, where United captain Scott Robertson and Paixao challenged in the air. One of them headed the ball into the roof of the net, with Paixao running away to celebrate.
The Portuguese winger almost doubled the lead when he ran onto Jon Routledge's diagonal ball and fired a bouncing volley just wide from 25 yards.
United got in behind their visitors for the first time when Russell sped away from Mensing, but Imrie read his low cross and flicked the ball to safety as Goodwillie prepared to finish.
The hosts continued to improve and got a break in the 42nd minute when a Douglas corner fell kindly for Goodwillie, who lashed the ball into the roof of the net from six yards.
Paixao, who was wearing a snood despite the International FA Board banning them with immediate effect last week, carved out the first chance of the second half but Antoine-Curier failed to connect properly with a header despite getting away from Timothy van der Meulen.
Mensing headed wide after another deep corner by Imrie before Goodwillie shot over from a tight angle after negotiating his way past three men.
Hamilton club captain Alex Neil, who made a brief but aborted comeback in October, came on for the injured Mark Carrington around the hour mark.
United began to push Hamilton back and brought on Buaben and Swanson in a bid to find some creativity with the former almost proving the difference with a 30-yard snap-shot that Tomas Cerny dived to push over.
Imrie moved from left-back to centre-forward following the introduction of Gavin Skelton but again a United substitute almost had an instant impact as Daly just failed to connect properly with a Douglas corner.
McAlister drove just wide from 20 yards at the other end and United had a huge let-off in the 79th minute when Van der Meulen brought down Imrie.
The Hamilton player looked at least a yard inside the box but referee Willie Collum awarded a free-kick.
Accies almost snatched a late winner when Routledge's first-time effort drifted just over the top corner following Imrie's cutback.
But they were undone when Goodwillie's neat reverse pass freed Daly to slot into the bottom corner from 14 yards.