Leeds' £3m winger Kemar Roofe scored his first goal for the club as they beat Wolves 1-0 at Molineux.
Roofe, a summer signing from Oxford, pounced in the 71st minute to earn Leeds only their second win in five games.
Leeds had to get the three points the hard way as they rode their luck in the opening period when Wolves were on top.
Wolves created a hatful of chances but paid the price for not making that dominance count and have now collected a solitary point from their last 15.
Following a low-key opening to the game, Leeds suddenly found themselves under a spell of intense pressure. A misplaced header by Eunan O'Kane after 18 minutes let in striker Nouha Dicko but he lobbed his shot over the bar from 12 yards while under pressure from central defender Pontus Jansson.
Having soaked up such a lot of pressure, Leeds badly needed to stop the flow and almost did it in spectacular fashion through Hadi Sacko in the 38th minute. A long ball out of defence by Kyle Bartley released Sacko, who saw his rasping drive fly to safety off the top of Carl Ikeme's crossbar.
It proved to be a brief respite for Leeds as less than 60 seconds later, they were defending frantically again with Jon Dadi Bodvarsson having his shot blocked before Teixeira's follow-up was saved by Green.
On the stroke of half-time, Wolves were again cursing Jansson as he deflected Costa's shot from another Teixeira cross over the bar.
When Costa again went close from long range in the 64th minute it seemed only a matter of time before Wolves' pressure finally paid off. But they were undone when right-back Matt Doherty lost possession cheaply. Sacko pounced on the loose ball and rolled a cross into the path of Roofe, who had the simple task of stabbing the ball home from six yards.
Leeds should then really have made the game safe three minutes from time only for striker Chris Wood to lash his half-volley over the bar from 10 yards.
That could have been a costly miss but Wolves' dismal afternoon in front of goal was summed up in stoppage time when Costa failed to hit the target at the far post from only six yards.
Wolves boss Walter Zenga:
"It is a little frustration with the situation. I think the team created and played well and tried to do the positive things. The negative point should be if the team doesn't play and doesn't create anything.
"The team played well and created chances. We have to convert these chances into goals. We have to continue to work hard, don't think everything is negative because the only negative is the result."
Leeds boss Garry Monk:
"Overall our character shone through today and we got the deserved three points. I think we saw it out quite comfortably. Three points and a clean sheet - you have got to be happy.
"I think that it was an excellent performance. A really gritty typical Championship game where you have to dig in. We showed that side of it and I think that part of control and bit of football that we want to put on the pitch was there in patches today."