Scott Arfield netted in the first minute to settle a contest which Leeds might have won on another day - especially if striker Chris Wood had taken some of the chances which came his way.
The towering New Zealand forward spurned two glorious second-half chances, sending a couple of gilt-edged headers wide of goalkeeper Tom Heaton's post as United slumped to a third defeat in four Championship outings.
Evans, bidding to secure a contract extension for next season, felt that his boys' second-half display against the Championship leaders was "fantastic", claiming that even some of the opposition were surprised to have snatched all three points.
"The performance in the second half was fantastic," Evans mentioned on BBC Sport.
"We've come here and been the better team by a distance, but we've lost the game. A number of their players said to me they've been outplayed today."
Indeed, it was much better from Leeds - especially given the fact that Burnley look like winning promotion this term - but there remains a distinct lack of killer instinct within the United side and it is undoubtedly harming both the team and Evans, whose future at Elland Road has not yet been clarified.