Mourinho was angry with the official for not awarding the Blues a second-half penalty when defender Craig Catchcart handled the ball as he blocked a shot by Diego Costa.
However, following Cathcart's intervention the ball immediately fell to Loic Remy who drilled home the Barclays Premier League leaders' second.
Friend later told Mourinho that he would have awarded a penalty had Remy missed and the explanation resulted in a personal apology from the Portuguese.
"I want to apologise for my earlier comments in some television (interviews)
What the referee did was great refereeing," Mourinho said.
"He saw that it was a penalty, he was going to give a penalty, but he saw the ball was going to Remy
And he waited a couple of seconds.
"He told me that if Remy didn't control the ball well or the ball goes over the bar or the ball is lost, he would have given the penalty.
"I was complaining, but it was good refereeing by Kevin
I didn't think it was a penalty, it was a penalty."
The half-time interval arrival of Willian and Costa was the catalyst for Chelsea to break the deadlock with the former, Remy and Kurt Zouma on target to secure passage into the fourth round.
Early in the second half Filipe Luis almost turned a Troy Deeney shot into his own net as Sky Bet Championship promotion hopefuls Watford pressed for the opener at a bitterly cold Stamford Bridge.
Chelsea made eight changes following their New Year's Day defeat to Tottenham and Mourinho admitted he had gambled in leaving Eden Hazard and Cesc Fabregas out of the squad, with Willian and Costa his insurance policy on the bench.
"I was worried at half-time because the game was always under control, but we weren't strong enough to score and when you don't score in these matches there is a risk," Mourinho said.
"We were always in control but the first half was a bit sad, so it was important for us to make changes and try to change the game
William and Diego did well.
"We had them on the bench because we knew it could be a risk and it was a risk
The first half was 0-0
I thought to win the game we needed to change some players and the system."
Mourinho announced that keeper Jamal Blackman had been recalled from his loan deal at Middlesbrough to cover for Mark Schwarzer's departure for Leicester.
Watford manager Slavisa Jokanovic admitted the introduction of Willian and Costa changed the game.
"We started to have more problems when Willian arrived
It was easier for Chelsea after they scored the first goal," Jokanovic said.
"In order for Chelsea to resolve this game, they had to bring on players worth millions of pounds
I can be happy with the performance, if not the result.
"We made some half chances during the game
In the second half Chelsea shot four times and scored three goals
That is a quality team."
Source : PA