Eto'o started in place of Ross Barkley at Turf Moor on Sunday in the only change to the side that beat Aston Villa last weekend - and the veteran forward made the most of his opportunity.
The former Chelsea striker headed home Leighton Baines' cross after only four minutes and sealed the points five minutes from time with a stunning long-range strike.
Danny Ings' first Premier League goal had drawn Burnley level following a mistake from Romelu Lukaku, but the Belgium striker made amends by putting the Merseyside club back in front before the break.
It was ex-Barcelona and Inter man Eto'o who took centre stage, though, demonstrating why he has played at the highest level for so many years to ensure Everton made it back-to-back wins in the top flight and ensure Burnley are still yet to secure a Premier League victory.
Everton manager Martinez was delighted with the contribution of the former Cameroon captain, which came just three days after he had played the full 90 minutes against Lille in the UEFA Europa League.
The Spaniard said: "I think everyone will talk about his goals and his finishes and I think it's enough to talk about those, because to have that clinical aspect about his play - as a striker it's not about how many goals you score, it's about how many chances you need to score.
"The first chance he showed that the mentality is to be ready, I thought he set standards in that respect, but then as a manager I was more pleased with his overall contribution.
"I was extremely proud of his performance."
Sunday's victory capped a fruitful week for Everton after they beat Villa 3-0 and secured a goalless draw in France.
And Martinez was impressed with the manner in which his side went about securing all three points against Sean Dyche's men.
"[It was an] Interesting week for us, it was a terrific challenge. To face a very difficult team at home against Aston Villa, then we had a really important game at Lille in the Europa League and then coming here today I knew it was going to be a really tough game," he added.
"I thought we were very, very professional. We controlled the game well."
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