The Clarets were by no means at their best but still did more than enough to beat the Millers, with goals from Sam Vokes and Scott Arfield in either half settling the clash at Turf Moor.
It could have been a different story entirely had the Millers not missed a golden chance to equalise with 10 minutes remaining.
Rivelin Valley Road Sheffield: Body found after police search of Rivelin Valley
Hosepipe ban: Reservoir near Sheffield less than 15 per cent full as photos show how water levels have plunged
The Long Blondes: Leadmill cancel gig after allegations surface against band creator Dorian Cox
Yorkshire Water announces hosepipe ban as Government prepares to declare drought in UK
Sheffield weather: Met Office issues thunderstorm warning as heatwave set to end with heavy rain and lightning
Joe Newell burst into the box but put his shot wide when it looked easier to score and then Arfield made the game safe six minutes later.
Warnock knew that miss was costly and said the end result would have been different had the chance fallen to a Burnley player.
"You are not going to get too many chances in the game like that," he said. "I felt second half we were lot more positive.
"Joe is disappointed because he could have just side-footed it in, he rushed it a little bit.
"It would have been a nervous end to the game, but it's not to be. Their lad (Andre) Gray would have just stroked it in really."
On his side's survival chances, Warnock added: "I am really happy, they can't be too down.
"You don't play Burnley every week. We have got some tough games but teams that we will play against will have some tough games.
"We are looking forward to the challenge."