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Both teams were also reduced to ten men in the second half with Aristote Nsiala and James Brophy recieving red cards for Shrewsbury and Swindon respectively.
Goddard, who signed in the summer from Woking for an undisclosed fee, netted a 54th-minute spot kick to give Town the lead, but it was not enough to get a much-needed win in what was billed as a six-pointer pre-match.
It was Luke Norris who was fouled in the box by Shrewsbury's Aristote Nsiala, who was sent off for his misdemeanour.
James Brophy too would get given his marching orders, two yellows for him though, in the space of a couple of minutes. The second one was for tripping Freddie Ladapo.
Swindon were enforced into two changes for this match after Darnell Furlong and Michael Doughty departed the club to return to Queens Park Rangers after their loans were ended early by the Hoops and their place in the line-up was taken up by Brad Barry and Sean Murray respectively.
Shrewsbury meanwhile handed debuts to all three of their January signings to date with Ladapo, Joe Riley and Alex Rodman all given starts.
The first half was devoid of quality as the opening 45 minutes only saw a smattering of shots with neither keeper, Lawrence Vigouroux or Jayson Leutwiler, having much trouble in their respective goals.
Jon Obika curled a shot over the bar with his weaker left foot early on and another effort was deflected into the path of Leutwiler.
Swindon's closest chance in the first period came on the 20-minute mark when Obika had a decent opening after being played in by Johnny Goddard but his shot was tipped over the bar by an advancing Leutwiler.
A corner followed and it dropped at the feet of Lloyd Jones but the centre back could only manage a shot that was blocked out for a corner that came to nothing.
Visitors Shrewsbury could not fashion anything threatening either though apart from a Mat Sadler looping header that needed tipping over the crossbar by Vigouroux.
Four minutes into the second half Raphael Branco headed a Sean Murray corner over the bar to give a small indication that the game was set to improve.
As soon as that had happened the Town End goal netting needed a lenghty repair from the referee and that was the catalyst for a barnstorming second half.
The goal kick that followed was poor and a pass sent Norris away who looked odds-on to score, especially after he had got around the goalkeeper, but he dallied in possession and while he was getting room for himself to shoot he was fouled in the box.
Goddard stepped up and netted the resulting spot-kick after Leutwiler dived to his right but the ball was caressed down the middle of the goal.
The hope was that the man advantage would give Swindon momentum to increase their lead and secure the points but with 20 minutes remaining the match was level in a numerical player sense when Brophy was sent off for two yellow cards.
And with ten minutes to go Swindon were punished for not seeing the game off when Sadler superbly guided a header off the underside of the bar and, apparently, in. Town hacked the ball away as the ball bounced on the line but the fourth official had already given the goal.
As soon as the goal went in, chants of "we want (chairman Lee) Power out" and "we want our Swindon back" came from sections of the County Ground crowd and booes followed the final whistle.
In terms of other match action, Jermaine Hylton had a second penalty shout for Swindon after a give-an-go move with Goddard but when the 23-year-old was tripped the referee waved the calls away.
Goddard was also one-on-one with the goalkeeper late in the second half but a sensational recovery tackle by Sadler denied him.
Swindon Town: Vigouroux; Barry, Rossi-Branco, Jones, Brophy; Rodgers (Kasim '72), Thomas, Murray (Hylton '78), Goddard; Norris, Obika
Unused subs: Henry, Spalding, Evans, Stewart, Delfouneso
Shrewsbury Town: Leutwiler, Riley, Sadler, Ogogo, Dodds, Brown, Deegan, Ladapo, Nsiala, Rodman (Whalley '77), Ed-Abd
Unused subs: Halstead, Grogan, Ebanks-Blake, Waring, O'Brien, Smith
Attendance: 6137 (259 away)
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