Substitute Jese fired home a late winner to earn Real a 3-2 victory over battling Valencia and keep them within five points of Primera Division pace-setters Barcelona and Atletico Madrid.
The striker scored with eight minutes left to give Madrid the spoils in a match which saw the managerless home side twice fight back from a goal down after Angel Di Maria (28) and Cristiano Ronaldo (40) had put the visitors ahead. Pablo Piatti (34) and Jeremy Mathieu (62) scored for Valencia.
"I am pleased with this first part of the season and I have high hopes for 2014," Ancelotti told the club's official website.
"The team keeps fighting, working hard and has a lot of quality. We have lost some points whilst looking to play the way we wanted. I think 2014 may be very good for Real Madrid.
"Five points aren't insignificant. (Sunday's) match was very important and we had a lot of pressure on us. We started this match eight points behind and it is not easy.
"At this point the table is clear. Barcelona and Atletico are doing very well. They have five points more than us and are the favourites right now, but I have high hopes for the second half of the season."
Valencia went into the match without a coach after Miroslav Djukic was sacked last Monday following a run of just two wins in nine league games and subsequent slide down the table. After Sunday's loss to Real, Los Che are now 11th, with interim boss and youth-team coach Nico Estevez left with a mountain to climb.
Ancelotti admitted the match had been a tough one in light of Valencia's current circumstances.
He said: "When a team changes the coach it has a lot of motivation and desire. The match was difficult but we did well overall. We needed to play with a high pace and after the first goal we dropped that a little.
"It was a good match and we had wanted to win until the last minute. We suffered but the result is very important."
Despite the win, the Italian knows he still has an area of weakness he needs to target in the new year.
"It is true we have let in a lot of goals from set-pieces," Ancelotti added.
"We've have conceded goals from dead-ball situations in the last two matches, but we have only conceded two (set-piece) goals in 26 matches. The last two matches were not good, but the other 24 were very good.
"Everything will go well with more concentration at set-pieces."
There was also an injury blow for Real to contend with as key midfielder Xabi Alonso suffered a perforated eardrum, apparently in a clash with Mathieu. Alonso completed the match but will undergo further tests before it is known whether he will miss any time.
Match-winner Jese, on a more positive note, described his vital goal as a "dream".
"This win is very important and to score was amazing," the 20-year-old said.
"It is a dream to score the deciding goal. We showed everything we like in the team and that we're a strong unit.
"The first half was not as strong but then the team reacted and went in search of the goal.
"I have scored in big matches in the Camp Nou and the Mestalla but I'm just happy with the win, which is very important for the team. We're doing well although we've dropped some points."