Cork completed his move to the Liberty Stadium on Friday for a fee in the region of £3m, and Monk admitted he was shocked to learn the 25-year-old was even available.
Swansea and Southampton meet at St Mary’s on Sunday but the deal was not completed in time for Cork to play against the team he turned down a new contract offer from.
Southampton confirmed on Friday evening that Cork had secured the move for an undisclosed fee.
Swansea boss Garry Monk said: “I pushed a little bit for him to come in the summer, but it didn’t happen because he was committed to Southampton and going forward with them.
“But when the opportunity came up again in this window, we were happy to have the chance to bring him in.
“I was surprised a little bit he was available. I understand an offer of a new contract was in place at Southampton, but for whatever reasons he felt he didn’t want to sign it.
“He wanted to come here and we have showed the same enthusiasm to sign him, so that’s a good recipe.”
Cork played nearly 130 games for Southampton after joining them in 2011 but he started only five Premier League matches since Ronald Koeman was appointed Saints manager last summer.
He has struggled to dislodge Morgan Schneiderlin, Victor Wanyama and Steve Davis from the Southampton midfielder but Monk feels he will fit in well in south Wales.
“He’s got good experience at this level and he is a British player which is good,” Monk added.
“He has a Chelsea background as well so he has got good pedigree, and he is at a good age to develop into a top midfielder.
“Jack will add to quality we already have at the club and my job is to mould him slightly into what we want to do here.”
Monk was also hoping to sign Norwich full-back Martin Olsson, but admitted a valuation was proving troublesome for the two clubs.
“We have our valuation of players and other clubs will have theirs of what players are worth,” he said. “If it doesn’t meet a deal won’t be done and that’s the same with every negotiation you go into.
“We put a bid in (for Olsson) but it doesn’t look as if the two clubs will agree on a valuation – and that’s where that deal is right now.”