An unfamiliar Cottagers side containing debutant Josh Passley and rookies such as Chris David and Muamer Tankovic were almost dumped out of the cup by their League Two opponents, who played the last 37 minutes with 10 men following Michael Doyle's off-the-ball kick.
Chris Porter's instinctive first-half finish had put the Blades ahead and that looked to be enough until Hugo Rodallega's score secured a 1-1 draw and a replay 15 minutes from time.
It may have been a different story had Meulensteen fielded a full-strength team but with a relegation battle to fight and a trip to Swansea on Tuesday night, that never crossed his mind.
"I think every person with a bit of common sense can understand (my team)," he said.
"If one person respects the FA Cup it's me and I'd love to stay in it but we play Sunday at 1pm for television and then go to Swansea on Tuesday then every man with a little common sense understands you have to maximise your possibilities and minimise your risk.
"That's what we are doing.
"The Swansea game is important to us so it works two ways: It gives a chance to the players who are going to face Swansea to have an extended preparation and a chance for the others to play here.
"That's what you've got a squad for. Every squad member is a valuable player and has to deliver when called upon."
Asked how a replay fitted his plans, Meulensteen said: "It's like an extended Christmas period for us. I haven't even taken my tree down yet."
Meulensteen was honest in his assessing his side's shortcomings on the day, admitting that Ryan Flynn might have been awarded a penalty when he collided with Aaron Hughes at 1-0 and admonishing Darren Bent for missing the target late on from a yard.
"I'm not happy with a replay, I don't think that is the right word because looking at the second half and the red card you should - as a Premier League team - be able to put the game to bed.
"We lacked a bit of quality. In hindsight we were a bit lucky to still be in the cup because one of the breaks they had could have hurt us and they had a penalty shout that could have been given.
"Darren's was a poor miss, especially for him because that's what he's made his life on - goalscoring and those little chances.
"That's where those players turn up and he's disappointed with himself as you can imagine."