Chelsea stopped the successive Premier League defeats as they ground down Hull at the KCOM Stadium.
Costa found the top corner with a similar strike to double the lead after Nemanja Matic’s powerful run.
Hull never recovered and could have been further behind but for Pedro’s poor finishing from six yards.
Cesar Azpilicueta, David Luiz and Gary Cahill made up the back-line with Victor Moses and Marcos Alonso providing the width and energy on the flanks.
At times the three became a back five to give Chelsea even more security and bodies in front of Thibaut Courtois.
Chelsea manager Antonio Conte promised “a solution” after his side went down with a whimper against Arsenal last weekend and he found his answer in a strategy he used at former club Juventus.
It did not initially appear to have stabilised a side who had conceded nine goals in their last four games in all competitions.
But the defence, along with the team as a whole, improved markedly after half-time and never looked like blemishing only their second clean sheet of the season.
At times it has been easy to forget that Victor Moses is still a Chelsea player but there is something about him that Antonio Conte likes.
After being loaned out for the last three seasons, Conte has chosen to keep Nigerian winger Moses at Stamford Bridge this season.
Victor Moses was busy up and down the right flang and Chelsea as a whole offered more threat down his side of the pitch
Playing as a right-sided wing-back, he was industrious and inventive and might have earned a first-half penalty when his trickery lured Hull forward Adama Diomande into a rash challenge.
Even the most ardent Hull fan will not have expected their season to continue in the way it started.
But equally they will be alarmed by how the Tigers’ form has flat-lined after wins over champions Leicester and Swansea in their first two games. They have taken just one point from their last five games and are three points above the relegation zone.
But the hosts’ defensive resilience wilted in the second half and the more they chased the game, they more vulnerable they looked.
The uncertainty over manager Mike Phelan’s contract is an unwelcome distraction for a side who will need to be fully focused to turn their scant resources into survival.
“It has been difficult to get forward but we’re not playing Mickey Mouse teams here. We have to be sensible with our approach and can’t just be open against these teams.
“My situation has fluctuated from week to week but we’re not far apart. We are talking and having lunches together, the owners and I. It’s just a case of where it all stands.
“Now we have to get together and move forward. We have a few more games now where we can hopefully be competitive. I don’t think there’s any issues with the players who are here, they are trying to win football matches.
Chelsea boss Antonio Conte in a statement said, “Every game is very tough in England and I am pleased with our performance because we scored two goal and got a clean sheet. I saw the right pressure and the right intensity from the players.
“After these two defeats it’s not easy to work because two defeats for Chelsea are heavy defeats. But this was a good answer.
“I saw many of the players with great commitment, attitude and will to change the situation. I am pleased for them. “We must follow this with work, work, work!”
Chelsea kept their first clean sheet in eight away Premier League games, since a 4-0 win at Aston Villa in April. Chelsea are unbeaten in their last 13 meetings with Hull, winning 11 and drawing two.
N’Golo Kante made more passes (79), more successful passes (70) and had more touches (95) against Hull than he had in any of his previous 43 Premier League matches.
Hull take the long trip to Bournemouth on Saturday, 15 October after the international break. Chelsea are at home to champions Leicester on the same day in the 12:30pm kick-off