Saturday's 2-1 win over Manchester United gave Monk's side their first league double over the 20-time champions, keeping them on course to challenge for a place in the top eight.
Swansea's highest finish since returning to the top flight came in the 2012-13 campaign when they finished ninth, the position they currently occupy in the Premier League, while the club's best placing of sixth was acheived 33 years ago.
"We have an opportunity to create a special season for ourselves," said Monk. "It may not be winning the league or avoiding relegation as it is right now, but we have different things that we have to fight for.
"Like I said to them before the game, you have to be remembered for something and we've got an opportunity this year to finish the season really strong.
"All that hard work that we've done up to this point has to be remembered and it's making sure that we finish strongly and don't just accept average.
"We need to make sure we push for better than that."
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