But goals from 19-year-old Barthram and on-loan Nicky Ajose either side of half time ensured 2012 runners-up Swindon kept alive their hopes of a Wembley return.
Blackman stunned the home fans with a deflected ninth minute opener but Barthram hit back just 10 minutes later, latching onto Grant Hall's pin-point cross to head home at the far post.
Swindon took the lead within a minute of the interval as Ajose claimed his third goal since arriving from Peterborough in August. The forward raced onto Bradley Pritchard's cute through ball before picking out the top corner from 18 yards.
The only downside for Swindon was the sight of Barthram being carried off on a stretcher mid-way through the second half and the final whistle means the Robins have now lost only once in eight home games - to Chelsea in the Capital One Cup.