United set up a trip to Pittodrie to face Aberdeen on Sunday when they beat Forfar to make it four consecutive wins and six games unbeaten.
They first take on a Kilmarnock side looking to get out of the bottom two in the Scottish Premiership.
Butcher said: "We have been on a good run and we want to keep that going. It's a really interesting end to the season with the cup games coming thick and fast and we want to finish as strongly as we can.
"It looks like we are safe but we want to pick up as many points as we can and keep that momentum going because we want to go into Sunday with confidence."
United drew at Rugby Park at the start of their unbeaten run before beating bottom club Hamilton in their most recent league encounter.
Butcher knows they will need to match their opponents' commitment first and foremost.
"They are tough games," the 30-year-old said. "When you miss out on the top six as we did by a point or so, then you are disappointed but you know you are going into the bottom six and teams are fighting for their lives.
"You have got to be prepared to roll your sleeves up and win that battle before anything else. We did that well against Hamilton at times and we need to do that against Killie as well."