Newcastle, without a permanent manager and with owner Mike Ashley still looking for a buyer for the club after relegation from the Barclays Premier League last season, looked to be holding out for the win until referee Chris Boyle awarded a second spot kick when Ryan Taylor appeared to man-handle Craig Conway.
Goodwillie made no mistake but there was a further moment of controversy when referee Boyle appeared to ask the Newcastle bench to replace the complaining Damien Duff but it all ended relatively peacefully.
The game was organised to mark the Tannadice club's 100 years of existence and the occasion was well supported by the couple of thousand Geordies who made the journey north of the border, many of whom were still outside when the game began.
Before kick-off there was a minute's applause as tribute to former England and Newcastle manager Sir Bobby Robson who died from cancer last week. For most of the game the Geordie fans alternated between singing in praise of Robson and against Ashley.
An indifferent start to the game was interrupted in the 10th minute when Newcastle midfielder Joey Barton finished off a move he started in the centre circle by volleying goalwards from 14 yards, only to see his shot blocked by Tannadice full-back Mihael Kovacevic.
Dundee United responded with a powerful Swanson volley from the edge of the box which Harper gathered at head height.
In contrast the mostly insipid football on show, there were some tasty tackles in evidence, notably Nolan on Tannadice stopper Andy Webster and Ryan Taylor on Terrors' striker Danny Cadamarteri which had the home fans in uproar.
The home side missed a good chance to open the scoring in the 50th minute when substitute Conway, on for Cadamarteri, drove in to the Newcastle box before setting up Prince Buaben with a perfect cut-back, then watched as the Tannadice midfielder side-footed the ball high over the bar.
But on the hour mark the visitors took the lead after Tannadice keeper Stevie Banks made a basic error.
When Duff's high and curling cross from the right-hand side of the box came over the back post, the former Hearts keeper failed to hold the ball as he crashed in to Kovacevic, leaving Carroll the easiest of tasks to slam the ball in from around three yards.
Five minutes later Nolan handled a Conway cross inside the visitors' box giving United the chance to equalise from the spot.
However, Swanson's penalty was well saved by Harper who dived to his left to parry before Terrors playmaker Morgaro Gomis blasted the rebound over the bar.
In the 71st minute Banks went some way to making amends for his costly error when he pulled off a fine save from Barton after the former Manchester City player had burst in to the Tannadice box.
But five minutes from time Dundee United were awarded a second penalty when Ryan Taylor was adjudged to have pulled at Conway inside the box and this time Goodwillie sent Harper the wrong way.
There was still time for Conway to blaze over the bar as the home side finished strongly, while Banks made another good save, this time from Nolan.