Alonso – who oversaw two victories in his first two matches in charge of Chelsea before failing to win any of the next three - gave Caicedo his first start of the season after being sidelined with a calf problem.
The Ecuadorian, 24, provided a stronger shield for Chelsea's defence that has looked shaky since the campaign began and he formed an effective partnership with 36 year-old Jordan Henderson who scored twice.
Alonso stressed the importance of Caicedo's return after a brief appearance in August when he lasted only 14 minutes as a substitute before having to limp off.
"With him we have a better midfield area and when you are better in the middle you have a more complete team and something to build around," the Spaniard told reporters.
"I am so happy that Moi is feeling better because he is going to be a key player for us. We knew that and we have seen (it) today."
In further encouragement for Chelsea fans, right back Marco Palestra - who signed from Atalanta in Italy in the close season but was injured soon after - made his debut as a substitute.
The two goals from Henderson and another two from Joao Pedro – both players who have also been in the medical room – and a Morgan Rogers strike underscored Chelsea’s reputation as one of the Premier League’s entertainers this season.
They added to their tally of early goals as they rushed into a 2-0 lead within 149 seconds and got their third a little more than 30 seconds after the start of the second half.
Cole Palmer celebrated signing a contract extension by providing two assists, adding to the signs that he is getting back to top form.
The only worries for Alonso were Chelsea's failure to keep a clean sheet for a 22nd Premier League game in a row when Bournemouth striker Evanilson briefly gave the visitors hope in the second half, and the absence of captain Reece James.
Alonso said James - who has missed long spells in recent seasons due to hamstring problems - had suffered a knee injury in training on Friday.
Chelsea will face stiffer tests this season than Bournemouth who have not won any of their six games so far under new manager Marco Rose and were missing key players including England midfielder Alex Scott and strikers Eli Kroupi and Justin Kluivert.
Alonso said Chelsea lost control of the game briefly when Bournemouth pulled a goal back through Evanilson to make it 3-1 in the 51st minute, but he took encouragement from the way they finished off the game with two goals in three minutes later on.
"From today we take positives for the future," he said.