Daniel Sturridge added a fourth three minutes after the re-start before Tomas Oravec scored a consolation goal in the 55th minute.
Ancelotti was particularly impressed by the Blues' first-half performance, but refused to get too carried away with his side's stunning start to the season.
"The first half was fantastic,'' he told Sky Sports. "We played very well on the counter-attack. We had a lot of chances. It was a fantastic performance.
"The second half was different. We had a good control of the game. We are doing very well. Zilina put more pressure on our defence. The back four did a good match. When you are 3-0 up it's normal to have more difficulty.
"The key was Anelka, he played a fantastic game. He created space, he had opportunities, he was the key for this game. He can play the position in front, sometimes we use him more wide, sometimes he plays in the middle but he shows the same skills if he plays wide.
"I hope to watch my team play a fantastic game for 90 minutes. Obviously I was a player so I know sometimes there are difficult times, the opponent can come back in the game. If I have to judge the performance I am very happy because it's not easy to win 4-1 away in the Champions League.
"We are in a good moment. We started well in the Premier League and in the Champions League - everything is okay now.''
On having achieved such a convincing win without the suspended Didier Drogba, the injured Frank Lampard and the rested Ashley Cole, he added: "We have an important squad. We want at this moment to use all the players.''
Chelsea have now scored 21 goals so far this season and conceded just two in all competitions but Ancelotti insists they are not looking to beat their 142-goal tally of last season.
"Our aim is to play good football, to win the game,'' he said. "We are not interested in beating the score from last year because it's our record, maybe we are not able to improve on our record.
After netting his first ever Champions League goal, Sturridge reflected on a job well done for the reigning Premier League champions.
"We played some good football in the first half hour,'' he said on Sky Sports. "We wanted to play the ball quickly and try to make sharp movements in the attacking half of the field and to defend well, and I think we did that quite well and it paid off for us in the end.
"Zilina were playing at home so they had the fans behind them and were attacking a lot but I think the defence played pretty well tonight. We were unlucky to concede a goal but we're going to come back stronger in the next game.
"We came out here to win and we're delighted to do that. I think everyone will be happy with the win because the main thing was to do that in the Champions League. It's a knockout competition to so get three points on the board is fantastic for the team.''
On his first Champions League strike, he added: "I've been dreaming of this since I was a young boy. I'm delighted to get off the mark in the Champions League and hopefully there's many more to come. There's a lot of world-class players in this team. I'm delighted to be part of things.''
Zilina coach Pavel Hapel was surprised by the scoreline and said the only difference between the two sides was Chelsea's clinical finishing.
"I am not disappointed with tonight's game,'' Hapel said. "We played well and attacked Chelsea. We managed to score so it was okay.
"We kept up with Chelsea as much as we could but I am disappointed with the productivity. The score was a surprise tonight. We didn't show the same productivity in front of goal as Chelsea did."