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The wide-open nature of the game was evident from the outset with Chris Wood missing an early chance, heading wide for the visitor's and then Ronaldo wasted a glorious opportunity, firing over the bar when clear on goal.
But it was not long before United had the lead -- Mason Greenwood's pass to Ronaldo was mis-controlled but Scott McTominay leapt on the loose ball and struck a low drive through the crowded area and into the bottom corner.
Luke Shaw then burst through the middle of the Burnley defence but fired his shot into the side netting as United visibly grew in confidence.
The second was not long in coming, after a mix-up in the Burnley defence, Jadon Sancho cut in from the left and his low shot took a deflection off the outstretched foot of Burnley skipper Ben Mee and beat Wayne Hennessey.
United, playing with an attacking 4-4-2 formation, were leaving space for Burnley to break into and looked far from secure at the back but their positive approach paid off when they grabbed the third goal in the 35th minute.
Hennessey tipped a fierce McTominay drive against the post but the ball fell straight to Ronaldo, who guided the ball into the unguarded goal.
That felt like game over but Burnley kept themselves in with a chance thanks to a fine solo effort from Aaron Lennon, the former Spurs winger taking advantage of a poor touch from Eric Bailly, dribbling into the United box and beating David De Gea with a well-placed angled shot.
Hennessey did well to keep out efforts from Greenwood and Cavani after the break while Burnley's chances were limited to headers at set-pieces from James Tarkowski.
With the win, United moved above Spurs into sixth, level on 31 points with fifth-placed West Ham United. Burnley remain in 18th place on 11 points.
Rangnick was pleased with the attacking performance in the first half in particular.
"The whole team performance was a lot better, I was pleased with the performances of our wingers. Jadon did well and Mason. Cavani and Cristiano, the work effort was amazing," he said.
United are four points behind fourth-placed Arsenal, with a game in hand, as they search for a way into the top four and a place in next season's Champions League.
"I wouldn't say fourth is the minimum. It is about taking next development steps. Now eight games in a row without a defeat. It is about taking next steps," said the German.
Burnley manager Sean Dyche said his team had paid the price for some lax defending.
"The frustrating thing for me is that we could have made two blocks on the edge of the box and we didn't," he said.
"I was quite pleased with the performance generally, we are a little bit stretched with COVID and injuries. There were some good performances but the details are killing us."