Jim McIntyre admits Dundee have a mountain to climb if they are to escape relegation from the Premiership.
Second-half goals from left-back Scott Tanser and striker Callum Hendry eased seventh-placed St Johnstone to their fourth Tayside derby win of the season.
But the ineffective Dark Blues, now without a win in their last eight games, have slipped six points behind closest rivals St Mirren with just 12 left to play for.
“It’s a major disappointment,” admitted McIntyre, who has to lift a team devoid of confidence for a must-win game with Motherwell next weekend.
“We’re not stupid. We know we need to probably win four on the bounce to still have a chance. That’s the reality of it,” said McIntyre.
“But we’ll keep trying. We’ll keep digging away. We’ve let 2,000 punters down today, who were giving us very good backing, but when the first goal goes in they vent their fury – and they’ve got every right.
“I fully understand where they’re coming from and we’ll take it squarely on the chin.”
Veteran striker Kenny Miller appeared to gesture to Dundee fans after being substituted but McIntyre said the decision was made in an attempt to kick start his team.
“We tried to make positive substitutions to try and stimulate things because people on the park weren’t performing,” he added.
“I make no apology for that. We were trying to get back into the game and trying something different.
“It was a really disappointing day because I expected far more after what I had seen in the previous week’s training.
“I thought in the first half we were okay without having quality going forward and we were fighting our corner. When the goal goes in it is about digging in.
“When you are on a run like this you can be fragile but you have to roll the sleeves up and get on with it.”
Delighted St Johnstone manager Tommy Wright welcomed a commanding display marked by an outstanding contribution by winger Matty Kennedy.
“It was a good, dominating performance and we got what we deserved. We could have scored more goals, especially in the first half, so I am really pleased with the attitude and quality of play,” he said.
“I questioned them at Kilmarnock and they have bounced back.
“We have got more points than last season and they should be pleased with how they have done this season because there has been an improvement.”