Anton Stach opened the scoring with a long-range strike in the eighth minute, Noah Okafor doubled the advantage from close range just after the hour, and Dominic Calvert-Lewin turned home a rebound four minutes later to put the game beyond a Burnley side who are already relegated and managerless following Scott Parker's departure earlier in the week. Loum Tchaouna pulled one back for the visitors in the 71st minute but it was a consolation.
Leeds United are on an unbeaten run of six and the result moved them ahead of Newcastle United in the table as they closed in on a second consecutive season in the top flight.
Farke was generous in his assessment of his players after the final whistle.
"It's a massive step forward for us. I am proud of the boys because to deliver at the crunch point of the season with the additional load of cup games, to hold the nerves and to take points from the last four Premier League games — this is the crunch period where everybody is normally nervous. To already be on 43 points at this stage feels amazing."
He singled out each of the goalscorers in turn.
On Stach, who has now scored five Premier League goals — four from outside the box: "Anton has this quality, also from the set pieces. He is a key player for us and we missed him a lot in the last weeks. It is a sign of the quality in the group that we have delivered without him."
On Okafor, who has become increasingly influential since arriving on loan: "His fitness level and workload has improved immensely, his work against the ball. He is a player with big potential and talent, which was always obvious to see throughout his career."
And on Calvert-Lewin, who claimed his 12th goal of the season: "Even in the last weeks, sometimes his goals from open play were not that impressive but I always liked his workload against the ball — his pressing ability to win headers, be physical and hold up the ball. Fantastic. The inspiration of the second goal with the backheel, he also deserves the goal. It's like a bottle of ketchup, all comes out at the same time."
Farke was measured when asked about survival being mathematically secured.
"We will celebrate once it is mathematically done, of course, but we will stay greedy for these types of performances."
Leeds host Tottenham Hotspur on 11 May.