Playing on the newly renamed Rafa Nadal court at the Real Club de Tenis, the Spaniard made light work of his Brazilian opponent to set up a tie against South African Kevin Anderson on Thursday.
The pair had met once before in the second round of the 2013 US Open, when Nadal won in straight sets, and his victory came nearly as easily here.
Nadal broke Dutra Silva's first serve, taking advantage of a double fault and flexing his muscles in the first set.
He lost his serve at the start of the second but quickly came back for the win, showing the form that led him to victory at the Monte Carlo Masters on Sunday.
Nadal said: "I want to thank everyone who makes this tournament possible, a historic tournament and one of the best of the year. It makes me enormously proud that this court bears my name. It's a huge honour and hugely satisfying.
"The court was a bit slower than usual because of the rain, but I'm happy, the (tournament) debut is always delicate. Tomorrow I will have to take another step forward against an opponent like Anderson. I'm happy to be in the third round.”
Earlier, Australian player Tomic pulled out because of a lower back injury before his clash with Murray, setting the Briton up for a clash in the next round with Spaniard Feliciano Lopez, who beat compatriot Albert Montanes 6-2 6-2.
World number one Murray entered the tournament on a wildcard after his third-round defeat by Spain's Albert Ramos-Violas at the Monte Carlo Masters last week.
He is trying to regain his form after his recent elbow injury problem, ahead of next month’s French Open.
Rain had delayed all the day's matches, but when the skies cleared David Goffin (Belgium) beat Nikoloz Basilashvili (Georgia) 7-5 6-0 and Chung Hyeon (South Korea) saw off Philipp Kohlschreiber (Germany) 6-3 6-4.
Read the original article on Eurosport: Nadal wins on own court to reach Barcelona Open third round