Atletico were dumped out of the Copa del Rey semi-final by city rivals Real Madrid on Wednesday, losing 5-0 on aggregate as they missed out on the chance to defend their crown.
The two defeats to Real came either side of a disappointing 2-0 loss at Almeria in La Liga, which subsequently saw them lose their place at the top of the table.
They bounced back strongly on Saturday though, posting a 3-0 home win over Real Valladolid.
And Costa – who scored their second goal – maintained that Atletico never lost belief.
"We knew this could happen," he said.
"People doubted (us) after three defeats, but in the locker room we did not. We played two games in quick succession against a great team like Madrid, but after that, there was no doubt.
"We went out to win and we did."
Atletico coach Diego Simeone praised his team's intensity from the start and was relieved they ended their losing run.
"The start was very intense with lines close together, good ball pressure after losing it and good ability to attack to get the game," Simeone said.
"The second half was handled in the best way and settled… (it is a) very nice situation is to have 60 points in the league.
"When you talk about strategy there are many hours of tension and concentration, many hours of work goes towards a situation and in truth it went very well. It was great.
"The second goal gives a good idea of the strengths and definition of Diego. Quick, handy and what I liked most was the intensity which the team showed."
