Poyet's men have won just one of their last 12 Premier League outings, that success coming against fierce rivals Newcastle United, and are without a point since the turn of the year.
As a result, Sunderland sit 16th - level on points with Burnley a place beneath them - and Poyet is keen for his players to step up over the coming weeks.
"We need to be the ones making things happen instead of waiting for a mistake, a refereeing decision or for the fans to take the team to another level," the Uruguayan told the club's official website.
"It is up to us and that is the mentality that we want the team to have on Saturday, so I'm sure we're going to be up for it and I would like to think we now have the opportunity to go and win three or four games in a row.
"It is an important week with an absolutely massive game on Saturday against Burnley."
Poyet could welcome back Adam Johnson and Wes Brown from calf and knee problems respectively this weekend, but Lee Cattermole (groin) remains a doubt and Jack Rodwell is suspended.
After a positive run either side of Christmas that saw draws at Manchester City and Newcastle as well as a victory over QPR, Burnley head to Sunderland having suffered back-to-back defeats.
Sean Dyche's men threw away two-goal leads to lose at Tottenham in the FA Cup and Crystal Palace in the league.
However, with Danny Ings having scored in each of his last three Premier League games, Dyche will hold out hope of picking up an important win and leapfrogging Sunderland.
Dyche will be without Matty Taylor (calf), Stephen Ward (ankle) and Kevin Long (knee), but remains convinced his tight-knit squad have what it takes to beat the drop.
"It's parked, it's gone, we move forwards," he explained in reference to the recent defeats.
"We like to play, we like the idea of the games spinning around. We had it last year and you get used to that.
"It's just finding that balance between rest and recuperation but also the work that needs to be done to make sure the players are topped up.
"We know how good they are physically. It'll probably be a surprise to people the amount of rest we do give them."
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