With games in the Premier League, FA Cup and Champions League in quick succession over the course of the last month, games have been coming thick and fast for the Blues.
Morata who was initially devoid of confident, regained his confidence and composure three minutes before the break when he opened the scoring for the visitors - his first goal since Boxing Day.
That tie was the only one of the four quarter-finals to take place though due to the weather, leaving seven teams still in the hat for Monday's semi-final draw, conducted by Smith live on BBC Radio 5 Live.
Morata finished well to open the scoring just before halftime after Riyad Mahrez lost possession and Willian instigated a swift counter-attack and played the striker in with a flawless through pass.
And his side responded from conceding to Jamie Vardy 14 minutes from the end of normal time to advance. Alvaro Morata ended a personal goal drought stretching back to December 26 as Spain forward put Chelsea ahead with a clinical finish in the 42nd minute.
"It was a very hard game for us. We are unlucky today".
But they will fancy their chances of reaching the FA Cup final for the second time in as many years under Conte after drawing EPL strugglers Southampton in the last four.
"This goal will be very important for the future for us".
"I'm a bit upset at the moment because I think we deserved another result", said the Frenchman.
"For Alvaro it was very important to score, but at the same time I am very pleased for his performance". He wants things to his way and right now they are not, so he can not hide his disappointment.
The consistency of Mauricio Pochettino's team since the turn of the year has seen Spurs build a five-point lead over Chelsea.
They will face Southampton for a place in the finals of the FA cup.
Twelve-time FA Cup champion Manchester United and Tottenham secured their passages to the semifinals on Saturday.
Chelsea's only truly challenging fixture in the remaining weeks of the season is the visit of Liverpool to Stamford Bridge in the penultimate game of the season.