The Dens Park outfit dropped out of the Clydesdale Bank Premier League after Sunday's 1-1 draw with Aberdeen.
But Saints passed up three opportunities of their own to demote the Dark Blues by failing to beat them two weeks ago and then drawing 3-3 with Hibernian ahead of Saturday's 3-0 defeat to Hearts.
A win from just one of those games was all that was required to secure Saints safety but their nervy displays saw Brown attempt to turn up the heat on Danny Lennon's men this week, even joking in a newspaper interview that the Buddies were going down like the Titanic.
But midfielder Newton, on-loan from Newcastle, told Press Association Sport: "It's been cat and dog with Dundee these last few weeks but they knew what they had to do to survive.
"It's been frustrating for us as we have not been able to get the win that would have put it all to bed and then there was the stuff in the papers.
"But we just put that out of our minds and focused on the football. We were aware of what had been said but we tried to be professional pay no attention to it."
Dundee's relegation confirms an eight successive season of top-flight football for the Paisley side but Newton admitted the Saints players had grown anxious as a 15-point lead over Dundee was cut to just seven following Brown's appointment in February..
"We're all delighted that we have finally secured safety," he said.
"It's been a bit nervous over the last few weeks with Dundee finishing strong and us not picking up points."