The Cameroon striker has left Russian club Anzhi Makhachkala to sign a one-year deal and be reunited with Mourinho, who was his boss at Inter Milan.
Eto'o is apparently 'plan b' after United rejected two Chelsea bids for Rooney, with Mourinho opting against lodging a third, having said he had back-up options.
The former Barcelona striker did not travel to Prague for Friday night's European Super Cup clash with Bayern Munich - neither did Willian, whose move from Anzhi was confirmed on Wednesday - but is eager to get into the action.
"I am very happy to be here but also anxious because I want to get playing as soon as possible," Eto'o said on chelseafc.com.
"It wasn't a hard decision. I saw the qualities Chelsea have, and I was very happy with Jose Mourinho before, so when the opportunity came, I was very happy to take it.
"I have had a career where I have been lucky enough to win a lot of trophies in different countries, but one of my last dreams as a professional is to come and play in England and see if I can have the same success as in my previous clubs.
"I don't see it as a final challenge, but to have a good time, enjoy my football, and have success with this team."
Eto'o, who will have to wait until after the international break to make his debut, with Everton away on September 14 the first opportunity, revealed he might have been a Chelsea player sooner.
But he opted to stay at Barcelona before moving to Inter as part of the deal which saw Zlatan Ibrahimovic move in the opposite direction.
"I've had a couple of opportunities to come to England before, including the first time Jose was here, but it didn't work out for one reason or another," Eto'o added.