Swansea backed up their midweek win at Arsenal by overcoming Norwich 1-0 to put nine points between themselves and the Barclays Premier League relegation zone
"It's a big statement as we've given ourselves some breathing space," said Curtis, again standing in for absent head coach Francesco Guidolin, who remains in a London hospital being treated for a chest infection.
"We're still only on 33 points and that is not going to get you safety.
"We know we've got a lot of work to do, I've told the players to enjoy the moment and come back next week prepared to work and make sure we get something from another massive game at Bournemouth.
"But we've given ourselves some breathing space and the most important thing is that we stay up.
"We worked awfully hard for this Premier League status and we don't want to give it up."
Gylfi Sigurdsson's 61st-minute strike - his sixth goal in 10 games - settled a scrappy Liberty Stadium contest which failed to spark into life until the Icelandic international finished from substitute Leroy Fer's pass.
"It was a poor game and both teams were caught up in the tension of the enormity of the game," said Curtis.
"They needed points like ourselves and it affected the play of both teams.
"We controlled the second half for long periods and got a goal from one of best moves of the game.
"We were hanging on a bit at the end, not necessarily due to good play by Norwich, but because we got tense
"They could have equalised at the end but we deserved it over the second 45 minutes.
"We've had a message from Francesco about how delighted he was
He's congratulated everybody and hopefully he will be with us next weekend."
Norwich remain in the bottom three on 24 points and have now lost eight of their last nine games.
But Newcastle's home defeat to Bournemouth and Sunderland's draw at Southampton means they are only one point from safety.
"Unless one of the teams above us has a horrendous run of form and doesn't win until now and the end of the season it [current points total] will not be enough," said Norwich manager Alex Neil.
"But they should pick up enough points to get over the line.
"We've got nine games left and the crumb of comfort is that the teams by us have only picked up one point today - and we've still got those two teams [to play] at home."
Norwich had late chances to level when Nathan Redmond fired wide and Patrick Bamford's near-post attempt was saved by Swansea goalkeeper Lukasz Fabianski.
"Both teams sensed it was a cagey affair, but I don't think Swansea will come off thinking they were better than us," said Neil.
"We had as many, if not more, chances, but for that 10-minute spell when they were on top they got a goal and we couldn't convert our chances.
"Both teams were resolute and looked defensively solid, but they scored the goal when they had the chance."
Source : PA
Source: PA