Hull need miracle as Burnley relegated from Premier League and QPR all but
Third-bottom Hull are two points adrift of safety after Saturday's 1-0 loss to Burnley, in which Danny Ings scored.
The East Yorkshire club's final two matches are a trip to Tottenham and against Manchester United at home.
"With points available, we have to believe we can get the results," said 54-year-old Bruce.
Hull dropped into the bottom three following Burnley's win at the KC Stadium and relegation rivals Sunderland's earlier victory at Everton.
It is their second successive defeat following Monday's home loss to Arsenal, which followed back-to-back wins against Crystal Palace and Liverpool.
Now they need a minimum of three points against Spurs or Manchester United to stand any chance of survival.
"No-one gave us a cat in hell's chance against Palace or Liverpool, but we believe we can get a result at Tottenham next week," added Bruce, who led Hull to promotion in 2013.
"We have got to play better - but we can't play any worse than we did against Burnley.
"Whether the occasion got to them, I'm not sure. Now we have got a mountain to climb."
Newcastle were the other big losers over the weekend despite ending a run of eight defeats with a battling draw at home to West Brom. John Calver's side came from behind to grab a vital point that gives them a 2 point cushion over Hull.
On paper, Newcastle have the easier final two games away to QPR (look like being relegated on Sunday unless they beat Manchester City) and at home to West Ham
Hull in contrast have to play Spurs away and Manchester United at home on the final day.
Leicester also face QPR and Sunderland in a vital six pointer.
But whatever the results elsewhere Hull know they need to win at least one of their last two games to stay up.