Mixu Paatelainen praised his Dundee United players' resolve as they beat St Johnstone 1-0, despite being reduced to 10 men for the last 30 minutes.
Bottom side United's first win in Perth in four years, and Kilmarnock's draw away to Partick Thistle, means the gap between them is down to five points.
"It was a wonderful effort. With 10 men, that spirit and character was there again," said the Finn.
"I can't praise the boys enough. They've got real mental toughness."
Ryan Dow had put United into a deserved lead midway through the first half.
However, Saints came back into the match and when Coll Donaldson was sent off for denying Chris Kane an obvious goal-scoring opportunity, the visiting fans may have feared another damaging day in their bid to avoid automatic relegation.
St Johnstone midfielder Murray Davidson hit the crossbar as the home side pressed for an equaliser but United held firm.
"We still have a long way to go but that was a small step towards it," continued Paatelainen.
"The mood is positive. Killie will pick up points so we need to be strong. They have a quality manager, they have fantastic players in every department, but days like this will help our chances of survival and that's what we hang on to.
"I'm really pleased with everybody's effort."
Saints manager Tommy Wright was frustrated at his team's lack of bite in the first quarter of the match.
"Up until they scored we didn't compete well enough, didn't make enough tackles, didn't win enough first headers or second headers," he lamented.
"I thought we got a slap in the face with the goal and we were much better after that. Probably they've defended well enough overall to win the game.
"When you give a team something to hold on to, even with 10 men, it makes it really difficult."
Alan Mannus had made two impressive saves prior to the United goal but he will have been disappointed to have allowed Dow's shot past him.
"I think in the build-up to the goal we should have done better," said Wright.
"We dropped too deep and we should have dealt with the cross better. He got a free contact inside the box.
"Because Alan was on the move, it hit the underside of his body and squirmed over the line."