The former Italy and Juventus chief guided his side to a 1-0 victory over Manchester United in Saturday's final at Wembley Stadium, 12 months on from lifting the Premier League crown.
Conte concedes that even two major honours in two seasons may not be enough to save his job, but Giroud - brought to West London by the Italian in January - hopes to see him stay put.
"It will be sad if the manager will leave," he told Sky Sports News. "We need to move on - I have got a World Cup and then I will think about next year. I want to win some more trophies with Chelsea."