Their performance and victory represented further signs of improvement following Monday's 3-0 defeat of West Brom that had come after successive losses to Watford and Bournemouth, leading to growing speculation surrounding Conte's position.
The west London side are now gearing up for the first leg of their crucial Champions League last 16 clash against the La Liga giants in west London on Wednesday night.
"We lost two games at the end of a busy period".
The Brazilian was the best player on the pitch by a mile on Friday evening and has undoubtedly put himself forward for a start against Ernesto Valverde's side on Tuesday night, with Conte conceding that he has "many good doubts" ahead of selecting his best team to face the Catalan giants.
And Blues boss Conte declared: "For sure his performance was impressive. At the same time, this type of game must give us great enthusiasm and we must be excited to play it because I consider Barcelona as one of the best teams in the world".
Cesc Fabregas sent Pedro away to make it 2-0 after 27 minutes and Willian then used his right foot to devastating effect to skim a low shot past keeper David Marshall.French worldwide Giroud clipped in a neat finish from Emerson's cut back to make the second half academic. Willian showed he is capable of scoring from range off either foot as his low drive cushioned the post on its way past Marhsall moments later for his 10th goal of the season.
The contest was arguably over just before the break, as Giroud capped off a standout first-half showing by securing his first goal for the club with a deft finish following some adroit work from debutant Emerson Palmieri on the left flank.
He told beIN SPORTS, "I told him 'no pressure, no pressure!'"
Many around Stamford Bridge might say that's been long coming, but it didn't take long for the second goal to arrive. "Its relieved me a little bit".
By then Hull had fluffed their chance at a consolation, Fàbregas' trip on Harry Wilson having earned the visitors a penalty.
Meyler turned around and ruffled Willian's hair - with both men all smiles - but the Hull star's tame spot-kick was then saved by Willy Caballero. And Willian was inches away from the hat-trick his display deserved when another long-range effort came back off the post two minutes from time.
Riyad Mahrez was back firing for Leicester as they beat Sheffield United to reach the last eight.