Furner was sacked yesterday with Leeds sitting third from bottom in Super League and having won only five of their 15 competitive games this season.
He is the club’s second coaching casualty in less than a year following long-serving Brian McDermott’s departure last July.
Watkins said: “It was a bit of a shock, to be honest.
“I got on with Dave really well. He’s a really hard-working guy, but it’s just one of those things that has happened.
“They have made a decision based on how we’ve been playing and performing. It hasn’t been good enough so they’ve made a decision to move on.
“Obviously it’s hard, it’s sad, but we’ve just got to try and move on from it.”
Asked if the players feel they have let Furner down, Watkins insisted: “Of course.
“I think looking back at the games we’ve played and the results and the performances, some of it has been, in patches, pretty good.
“But there’s been areas where we’ve needed to improve in terms of managing the game a bit better and we haven’t done that.
“We haven’t come up with the performances we needed to and haven’t got the results we needed so of course we take responsibility for what’s happened.”
Richard Agar has stepped up from temporary assistant coach to become interim team boss.
He said: “My thoughts are definitely with Dave.
“It is a very, very difficult time for everyone, but in the short term we have just got to get on with it.”