Daly netted a bizarre opener at Fir Park after a slip by Motherwell goalkeeper Michael Fraser and brilliantly volleyed home after evading his marker at a corner just before the break.
John Sutton had levelled in between and headed home again with nine minutes left following Craig Conway's goal, but United withstood some late pressure and several penalty claims to secure a top-four finish.
United made six changes following the midweek draw with Rangers but they again showed their strength in depth as they extended their unbeaten run to 11 games.
They had a major hand though with Fraser gifting them a fifth-minute lead. The goalkeeper had plenty of time to clear after collecting a passback from Stephen Craigan on the edge of his box, but he lost his footing and the ball Daly and ricocheted into the net as he scrambled back to his feet.
The patch of grass was one of the small areas of the pitch that was re-turfed in January but it had been watered before kick-off.
Motherwell had some decent possession but were lacking a cutting edge and they were grateful for Goodwillie slipping in the same area as Fraser when Swanson played the ball over the top.
Mark Reynolds had a great chance to level following a half-cleared corner midway through the half when Tom Hateley's pass found him unmarked 16 yards out, but the defender shot inches wide of the far post.
The hosts forced several more corners and Sutton took advantage of Hateley's delivery in the 28th minute to power home a header at the back post following a good run and leap. There was no sign of the opening goal affecting Fraser's confidence when he sidestepped Goodwillie inside the box and cleared after collecting another passback.
Motherwell, who were without Keith Lasley and started Lukas Jutkiewicz on the bench amid his recurring knee problem, lost Craigan to injury in the 37th minute. Winger Chris Humphrey came on at right-back with Steven Saunders moving inside.
United regained the lead in the 43rd minute when Daly escaped his marker and expertly volleyed home Conway's corner, which was needlessly conceded when Reynolds miscontrolled Fraser's poor pass.
Motherwell had an instant chance to level when Jamie Murphy cut the ball back for Giles Coke, but the midfielder fluffed his shot and both Goodwillie and Daly threatened in injury-time.
Conway extended United's lead in the 49th minute. The winger broke down the left after Humphrey slipped and ran 40 yards before cutting inside and firing a shot that went straight through Fraser and into the corner.
Motherwell were struggling to get back into the game but they had two decent chances from set-pieces as the game approached the final quarter. Dusan Pernis blocked Coke's close-range effort after Jim O'Brien's free-kick and Reynolds was too high with a free header from Hateley's corner.
Motherwell then had three penalty appeals rejected by Calum Murray in quick succession, the first coming when Pernis barged into Sutton following Humphrey's cross. Substitute Jutkiewicz and Saunders then claimed they were held by Darren Dods.
Motherwell's pressure paid off in the 81st minute when Sutton headed home after Hateley had stood up a cross.
The home side pushed forward but could not find a way through as United, who brought on Scott Robertson for his first appearance since December, moved seven points behind Celtic to keep their slim Champions League hopes alive.