Boro striker Fletcher opened the scoring with a low, curling strike on 10 minutes.
But Callum Wright and Daniel Butterworth gave Blackburn a 2-1 lead going into half-time.
Marvin Johnson made it 2-2 from a narrow angle, Scott Wharton headed Rovers back into the lead before Fletcher struck again on 62 minutes.
Middlesbrough are above second-placed Reading on goal difference, with a match in hand, while Rovers are in third, two points behind Boro having played one match fewer.
Middlesbrough: 19 Jan v Nott'm Forest (H), PL Cup
Blackburn: 22 Jan v Cardiff (A), PL Cup