Ritchie, Raffaele De Vita and Miles Storey all scored before the break for the Robins - who also missed a penalty through Tommy Miller - while Adam Rooney added a fourth in the second half.
Swindon took the initiative from the off and went ahead in the 17th minute. Richie nodded Storey's deep cross into the path of De Vita, who took one touch before unleashing an eye-catching volley past Boro goalkeeper Steve Arnold.
On the half-hour mark it was 2-0 when Richie tried his luck from long range and his strike somehow eluded Arnold.
Full-back Alan McCormack was then fouled inside the box but Miller's penalty was brilliantly palmed around the post by Arnold. But it mattered not, as Storey met Ritchie's cross and guided a fine header home after 38 minutes.
And the visitors were not done there - substitute Rooney collected John Bostock's pass before calmly slotting home on the hour to seal a resounding victory.