Their fourth derby defeat of the season - and sixth consecutive loss in the SPFL Premiership - meant Hibs are now one point above 11th place with only three games left. They are at home to Partick Thistle on Saturday then face Ross County away and Kilmarnock at home.
Dreadful defending allowed Callum Paterson to convert two headers before Jordon Forster scored Hibs' first goal in over four hours of play. Despite some spirited play, and chances, they crashed to another loss.
Encouragement came from an unexpected source, though, as Locke praised them after what could be the last Edinburgh derby for some time. "They [the Hibs players] gave Terry [Butcher] everything today and I fully expect them to stay up," he said. "They put us under a lot of pressure and made it very difficult. If they play with that spirit and commitment in the next couple of games they will win them.
"I'm over the moon. We have been on a decent run of late and it was great to keep it going. We have been playing some good football but they had to show a different side to them today. We had to battle. It was very physical and I was really proud of them. We've won four out of five derbies this season which is a great achievement but I'm really disappointed at going down [to the Championship]. I'd swap the derby wins for staying in the league."
Hibs had fewer than 11,000 fans at the derby (Hearts took 3845) and some were so disgusted they left when Paterson's second went in after 41 minutes. Later Butcher appeared more relaxed, and philosophical, than many in the stands. "It was close but that's no consolation when you lose a derby match. That's the best we've played for a long, long time.
"I can't remember the last time we played as well as that. I felt we didn't deserve to lose the game but we have shot ourselves in the foot really.
"We've lost the game ourselves with two poor pieces of marking. Paterson's got two free headers and it's 2-0 and all that good work and everything else is undone by that.
"All I'm saying now is that if we take that quality of performance and that fight and determination into the next three matches then we'll be fine. If we can play like we did today, under that pressure in a derby match, when they knocked it about well, tested Hearts all the way, with the players taking responsibility on the ball and showing a lot of bottle and a lot of fight, I think we will come through these next three matches very well.
"I've been in this position as a player and a manager before. You need to hold on to and grab on to things that are positive and I think there were lots of things today that were positive. We looked as though we could carve a team open."
Forster accepted responsibility for poor defending at the crosses for Paterson's goals: "I definitely could have done better for the two Hearts goals," he said.
"But as a team we need to do better, stop giving away silly free kicks, defend a lot better as a team. The fans have got a right to be worried."