Federer, twice a winner on the Miami hardcourts, needed just 62 minutes on a sweltering centre court to dispose of Dutch qualifier Thiemo De Bakker 6-3 6-3 while Murray appeared to be adjusting just fine to life without coach Ivan Lendl, easing past Spain's Feliciano Lopez 6-4 6-1.
After getting the defence of his Sony Open title off to a shaky start with a three-sets win over Matthew Ebden on Friday, Wimbledon champion Murray was more in command against Lopez improving his record to 9-0 against the Spaniard in just 73 minutes.
"It's not always that easy to feel comfortable against him because there is not loads of rhythm with the way he plays," said the sixth seeded Scotsman. "But I moved well; returned well.
"It was obviously a more comfortable score line than the other day."
Murray has been in the Miami spotlight for more than his play this week after announcing he had ended his successful partnership with Lendl, who helped steer him to a pair of grand slam titles and an Olympic gold medal.
Lendl, who maintains a Florida residence, was in the stand catching some of the Sunday match, leading Murray to joke that the two men were renewing their partnership.
Seven of Murray's previous eight matches had gone three sets but the Briton, who has a Miami residence, made sure he would spend as little time as possible under a punishing sun, breaking Lopez three times in the second set, sweeping the last four games.
"It was really hot. I don't know how hot it was but when we came over at like 9:30 it was already 80 Fahrenheit (27 Celsius) and it was extremely humid," said Murray. "I have played a few matches here where it's been very, very hot but it doesn't always stay like this for a few days in a row in March.
"It's not easy conditions."
A resurgent Federer has shown that he is coming to grips with a new larger racquet, a change that the Swiss had resisted.
But Federer has seen improved results with the racquet, collecting his 78th career title in Dubai in February and reaching the final in Indian Wells to move back into the top five of the world rankings.
"I think it (the racquet) just gives me easier power, especially on the first serve," said Federer. "And I think as time now goes by, I'm getting a better feel for how is the ball going to react, you know, how much spin do I need to put on to stay really accurate and put it really close to the line.
"But then again this wasn't Andre Agassi on the other side, or Djokovic. It's just important for me to keep that up and remind myself it's going well so far."
In other third round action, Frenchmen Richard Gasquet and Jo-Wilfried Tsonga advanced in contrasting fashion.
The ninth seeded Gasquet had easy passage beating 17th seeded South African Kevin Anderson 6-3 6-4 while 11th seeded Tsonga rallied for a 4-6 7-6(6) 7-5 win over Cypriot wildcard Marcos Baghdatis.
Spaniard Tommy Robredo, the 16th seed, advanced with a 6-3 6-4 win over Frenchman Julien Benneteau.
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- Feliciano Lopez
- Ivan Lendl