The Frenchman then added a third to seal the win in injury time.
It was Chelsea's first game since the sacking of Luiz Felipe Scolari and Anelka chose to concentrate on the win rather than the circumstances surrounding the Brazilian's departure and the subsequent appointment of Guus Hiddink.
"I don't know what he (Hiddink) will learn from that performance," Anelka told Setanta Sports.
"Maybe he will learn that we are strong team.
"I don't know if it was a good change. You'll have to speak to the board about it, anyway the most important thing today is not about the manager, it was the result and the win."
Watford manager Brendan Rodgers admitted he was sad to see his side lose after going ahead, but insisted he could not have asked for anything more from his players.
"We have given a world-class squad a fantastic game," Rogers said.
"We were excellent tonight. With 20 minutes to go, we were 1-0 up and looking comfortable.
"I couldn't have asked for any more from my team.
"When we went 1-0 up I thought we looked reasonably confident. The boys have worked really hard and they were really disciplined today."
"I was trying to get the attention of the referee before we conceded to try and get a sub on but his ear-piece wasn't working so that was a bit disappointing but that's how things go.
"It was a great ball in for the first goal and Jobi (McAnuff) could have equalised for us at the end as well."