Swindon went on to win 3-0 and eased their own relegation worries, but it was the hotly disputed nature of their second goal, immediately after half-time, that effectively decided the outcome.
Simon Cox, who had headed Swindon into a 29th minute lead, appeared to be standing yards offside in front of goalkeeper Peter Gulacsi, when Owain Tudur-Jones played the ball back into the Hereford six-yard box.
Cox flicked the ball past the Hungarian keeper, playing his 9th match on loan from Liverpool, to score his 24th goal of the season and his fourth in the past five days.
Hereford, facing a fifth successive defeat, protested furiously that Swindon's top scorer, and two team-mates, were yards offside.
But referee Attwell, involved in a hotly controversial Reading goal at Watford, last Autumn and who sent off two Premiership players in West Ham United's recent visit to Wigan Athletic, awarded the goal after lengthy discussions with his assistant, Richard Kendall.
The goal completely knocked the stuffing out of Hereford, who are without a goal in four games and who sank to the League One basement as a result of this defeat.
Away from the controversy, Swindon thoroughly deserved a second consecutive win, which has lifted them seven points clear of the danger zone.
Cushioned by Cox's 29th minute header, they controlled large periods of the match and were worth the three points.
Febian Brandy, on loan from Manchester United, was a bundle of skill and energy for Hereford but was basically the visitors' lone threat.
The equally impressive Hal Robson-Kanu struck an upright for Swindon, who sealed their victory 14 minutes from time.
Appropriately Robson-Kanu was the creator of the goal, which Billy Paynter struck past Gulacsi into the far corner.