Poyet's side needed just a draw against West Brom on Wednesday to effectively move out of the reach of Norwich City due to their superior goal difference, but they went even further with a 2-0 win at the Stadium of Light.
Colback, who had last scored against Newcastle United in January, got the hosts off to a flyer as he netted after just 13 minutes and a second was not far behind.
On-loan Liverpool forward Fabio Borini doubled their lead just after the half-hour mark and West Brom – who themselves are effectively safe – were unable to mount much of a response.
With Sunderland staying in the top flight for at least another year at the expense of Norwich, Colback was quick to highlight the importance of Poyet in steering them to safety.
"Gus Poyet has been brilliant since he came in," the 24-year-old told BBC Sport. "He is positive and gives us belief. We had spells where we struggled but we've managed to pull it out of the bag somehow.
"After the Tottenham and Everton games everyone wrote us off but I think that helped us. To get the win tonight makes it feel good.
"I feel relief. It's been a long, tough season."
Captain John O'Shea acknowledged that the club looked to be in severe trouble after a run of seven league matches without a win followed their League Cup final defeat to Manchester City, but the former Manchester United defender praised their team spirit.
He added: "We stuck together as a group and, thankfully, the last few weeks we have gone on an amazing run.
"After the League Cup final defeat we had a lull and everyone thought we were in trouble but we took care of things here."
Liverpool report: Potential Ibrahima Konate replacement gives go-ahead for transfer
'I think that’s the respectful thing for two captains to do after a game. I probably boiled over a bit at the end, with a bit of everything coming out': Jarrod Bowen lifts lid on heated clash with Mario Lemina after West Ham's win over Wolves
