Spain striker Morata broke the deadlock with a well-taken finish in the 42nd minute as last season's beaten finalists prevailed 2-1 in extra time.
It was his first goal in 2018, marking a successful return to the starting XI after being overlooked for the last three matches.
The 25-year-old came close to a second on a couple of occasions and won praise for his all-around display.
"For Alvaro, it was important to score but I am pleased for his performance," Conte told reporters.
"He showed great character and to be strong. I think his goal will be very important for the future."
Conte, however, rejected the idea of being in an enviable position.
"I think that we must be ready to fight to reach the final and it will be very important for us. Southampton has the same idea.
"Last season it was a pity to lose the final and not celebrate the double. Our target is to do our best. If we can reach the final I want to change the result."