Dyche's side looked set for a valuable point when Beto's 87th minute goal cancelled out Dominic Solanke's opener at the Vitality Stadium.
But in the final seconds, Adam Smith's cross hit Everton defender Seamus Coleman on the chest before dropping into the net.
Everton sit just three points above the relegation zone with nine games left in their latest battle to beat the drop.
With a potential second points deduction for alleged financial breaches also hanging over them, Everton face a nervous end to the season but Dyche remains confident.
"The question marks over this club have been here for two-and-a-half, three years," he said.
"I believe in the players and I believe in myself."
Everton expect the verdict on a potential punishment to be delivered soon after they had a 10-point deduction for separate offences reduced to six on appeal.
Next up for the Toffees is a trip to Newcastle followed by a clash with relegation rivals Burnley at Goodison Park.
"Obviously I'm frustrated and disappointed with the dying embers of the game," Dyche said.
"For such a simple moment we get confused and they get a goal out of nothing really.
"We find a goal and I'm pleased with that, because it's an ugly goal and you can't always score the perfect goal.
"We should go away with a point of course, but we can't defend like that."
Bournemouth took the lead in the 64th minute as Solanke got between James Tarkowski and Ben Godfrey to head home Lloyd Kelly's cross.
Everton hit back when Bournemouth goalkeeper Neto dropped Dwight McNeil's cross right at the feet of Beto for a simple tap-in.
But Dyche's delight turned to despair after Coleman's own goal sent Everton home empty-handed.
