But four goals in 13 minutes after the break from Downing (two), Emile Heskey and Gabriel Agbonlahor kept Villa on course for a Champions League spot before Martin Paterson's late consolation.
In contrast, Burnley have now lost six of their last seven league games.
It was just the tonic Martin O'Neill's side needed at the start of a massive week with an FA Cup fifth-round replay against Crystal Palace on Wednesday and the Carling Cup final next Sunday against Manchester United at Wembley.
Villa had only scored three goals in their previous seven league games but still carried too much firepower for Brian Laws' side, who remain in the bottom three.
Burnley have still only taken one point on their travels and have now conceded 43 goals in the process.
Fit-again Agbonlahor teed up Villa skipper Stiliyan Petrov for the home side's first chance on the edge of the box but he dragged his shot well wide.
Then Burnley stunned Villa by taking the lead through Fletcher's 11th goal of the campaign.
Jack Cork's pass allowed David Nugent to break into the Villa box and his low centre was turned home at the far post by Fletcher, who had managed to remain on-side.
It was Burnley's first away league goal in 550 minutes since Chris Eagles scored in the 5-3 defeat at West Ham on November 28.
Nugent made a powerful burst towards the Villa area and his shot took a deflection which had goalkeeper Brad Friedel scrambling across his line before it drifted just wide.
It was 24 minutes before Villa created their first clear-cut opening.
Downing delivered a probing centre from the right and Heskey flashed a glancing header only a couple of feet wide of the far post.
Villa finally drew level after 33 minutes through Ashley Young's seventh goal of the campaign.
Young played a short corner to Milner, received the return pass and then cut inside before firing a ball into the box which clipped Daniel Fox before entering the net with Jensen unsighted.
Jensen then got his hand to a dipping shot from Ashley Young and the rebound hit Downing but flew wide.
Burnley skipper Clarke Carlisle became the first player to be booked after 40 minutes for a late challenge on Ashley Young.
Villa took total command after the break and went ahead in the 56th minute after a quick break down the left involving Heskey and Agbonlahor.
Ashley Young eventually sent over a low cross and Downing drilled an angled shot into the corner of the net.
Three minutes later the former Middlesbrough player added another. Jensen beat out a shot from Agbonlahor but Milner responded quickest to the rebound, turned the ball back inside to Downing and his shot deflected past Jensen.
Then in Villa's next attack Heskey got on the scoresheet when he diverted a low centre from Agbonlahor into the corner of the net.
Burnley were now in total disarray and after 69 minutes Agbonlahor completed Villa's nap hand. Heskey found Milner in space and Agbonlahor prodded the ball home despite being under pressure from Carlisle.
Agbonlahor and Heskey left the pitch to standing ovations when replaced by John Carew and Nathan Delfouneso respectively.
Fletcher was yellow-carded shortly afterwards for a foul on Villa left-back Stephen Warnock.
In a rare Burnley attack Friedel dealt with a well-struck shot from Eagles but Villa's quickfire scoring burst had decided the contest.
Clarets substitute Frederic Nimani was booked for deliberate handball before Carew could have made it 6-1 but he dragged his close-range shot wide.
In injury-time, Paterson pulled one back for Burnley from Cork's low centre.