Sutton set the tone for an entertaining quarter-final when he headed home in the first minute, but Goodwillie stunned the visitors with an acrobatic effort just before the break.
Motherwell had barely stopped celebrating Sutton's second when substitute Daly forced a replay in the 73rd minute.
Another match is the last thing United manager Peter Houston would have wanted before the game amid their heavy fixture schedule, but Motherwell came closer to averting the need for the March 29 replay in the closing stages.
The visitors netted after 50 seconds when Stevie Hammell swung a corner towards the back post and Sutton timed his jump perfectly to head down and into the net from six yards, but Goodwillie levelled in spectacular fashion in the 40th minute when he produced a superb overhead kick to beat Shaun Hutchinson to the ball eight yards out.
Sutton converted from close range at the near post to put Motherwell back in front, but United levelled after Swanson got to the byline on the right and lifted the ball across the face of goal. The cross was too high for Goodwillie, but Daly emerged unmarked at the back post and slotted home with his left foot from eight yards.
Source: PA
Source: PA