Goals from Jack O’Connell and Mark Duffy left the Millers – who had captain Will Vaulks red-carded on 28 minutes – in the Championship drop zone.
They are just one point adrift of fourth-from-bottom Millwall with 10 games remaining, but three points behind Wigan and Reading.
Warne knows he cannot afford that gap to get any wider before the next international break, with a trip to QPR on Wednesday followed by Saturday’s visit of leaders Norwich City.
“For the next two games I think we need to bring something back from our travels midweek and then we are always good at home,” said Warne. “I have told the lads how important the Norwich game is.
“When we go into the international break ideally we will be three points clear of the bottom. As long as we are in touching distance we are doing well.”
Rotherham arrived at Bramall Lane unbeaten in six outings, but the first-half dismissal of Vaulks, who will serve a three-match ban, crushed any hopes of extending that sequence.
Warne said: “We’ve been on a good run and even with 10 men my lads gave everything. That’s all I can ask. Sheffield United are a really good side and we frustrated them and, in the first half, I thought we were causing them problems.
“We will just keep ourselves together and then we will go again against QPR and Norwich, then it’s the international break. I’m not dwelling on this.
“Sheffield United are one of the best, if not the best home team in the division. “There’s no shock or shame that we have lost here.
“The sending-off isn’t an excuse, but clearly, every team is better with 11 players rather than 10 – especially us. We coach the lads well and they try to be the best they can be. Minus the sending off, they probably were the best they could be.
“There’s no disappointment in that. If it isn’t enough at the end of the season then that’s my head on the block, but the lads have done everything they can.
“We’ll go into the international break with a smile on our faces no matter what. I’ve had better days but I’ve had worse. I’m proud to be their manager.”
Three points for Sheffield United keeps them in touching distance of top two Norwich and Leeds United.
Victory also virtually assured them of the minimum of a play-off spot as they sit 14 points clear of seventh-placed Bristol City.
The Blades welcome Brentford to Bramall Lane tomorrow, having not conceded a league goal at home in 2019, before a crunch trip to Elland Road on Saturday.
Manager Chris Wilder has urged his players not to be distracted by the trip to Leeds, who won 1-0 at Bristol City, and focus on tomorrow’s visitors, who triumphed 2-1 at Middlesbrough on Saturday.
“We’ve got to get to the international break, get through this big week, then take a deep breath and steam into the last eight games,” said Wilder.
“The Championship can hurt you. Everybody looks at Leeds, but it won’t matter if we don’t get Tuesday right.
“It’s about getting through the race and the three who go up will be the three best equipped. We only look at ourselves, it’s only natural people talk about putting pressure on others.
“But we just try to prepare as best we can and then move on. That’s how it will be. If we’re in and amongst it with three or four games to go then great. We’ve just got to keep grinding away, which is why this was a big result.”
Head coach Marcelo Bielsa said Leeds United were enjoying the pressure of the tense automatic promotion race after keeping their place in the top two with a gutsy success at Bristol City.
Leeds held their nerve to complete a clean sweep of victories for the top three as Patrick Bamford’s ninth-minute strike sealed a 1-0 triumph at Ashton Gate.
Bamford collided with the post in netting a close-range finish with the striker eventually substituted in the 57th minute as part of a game in which Whites midfielder Kalvin Phillips was also hampered by a twisted ankle, but played through to the final whistle.
Bamford sat in the dugout with ice on his leg during the closing stages and is set to undergo scans on his knee though Bielsa is hopeful both the striker and Phillips will be fit for Tuesday’s game at Reading.
“It’s very difficult to use the word ‘enjoy’ after a game, even if we have won,” said Bielsa.
“But after, when we watch, when we have had an analysis sometime after the game, we feel we are enjoying this situation.”