The forward, who was recalled to the team after recently being linked with a switch to the Chinese Super League, opened the scoring in added-on time in the first half before Gary Cahill netted a second late in the game.
The win - Chelsea's 15th in 16 matches in the top flight - takes them eight points clear at the top of the table, while Hull remain in the bottom three despite a spirited performance at the home of the league leaders.
Costa almost announced his return with a goal after just 11 seconds but after the ball fell invitingly to him on the edge of the penalty area, he fired his volley just wide of the post.
Hull could have sat back from the off, but to their credit, they fashioned an opening in the ninth minute when Tom Huddlestone played a neat one-two on the edge of the penalty area before firing wide of the far post.
Soon afterwards, the game was hit by a nine-minute delay after Cahill and Ryan Mason clashed heads, but while Cahill was declared fit to continue, Mason required extensive treatment before being stretchered off.
Once the game got back underway, it was Hull who made the first move with Harry Maguire heading straight at Thibaut Courtois after he had initially won a corner with a deflected effort from an acute angle.
Chelsea gradually applied more pressure but they were limited to a shot from distance from Marcos Alonso, while Maguire remained intent on taking the attack to Chelsea with a long-range strike of his own which was tipped wide by Courtois.
However, the Tigers could not hold on until the break and in the seventh minute of added-on time, Costa found the bottom corner from eight yards out after meeting a cross from Victor Moses.
Five minutes into the second half, the visitors were denied a clear penalty when Marcos Alonso kicked Abel Hernandez, but both the referee and his assistant - who was standing directly next to the incident - waved play on.
It was a harsh call on a Hull side who were impressing in West London, and it was not long before David Meyler warmed the hands of Courtois with a fierce strike from 25 yards.
However, for all their pressure, only a Michael Dawson effort tested Courtois and with nine minutes left, Chelsea netted their second through Cahill, who ghosted through a group of defenders to head in Cesc Fabregas's free kick from close range.
Soon afterwards, Costa was denied a second by Jakupovic, while up the other end, Oumar Niaase forced an excellent save from Courtois after a volley on the turn.
Chelsea had little problem in seeing out the rest of the match, however, to strengthen their position at the top of the standings and leave Hull in 19th place.