Carles Puyol and Pedro also scored to as Barca claimed an immediate lead in the standings over arch-rivals Real Madrid who were held 1-1 at home by a feisty Valencia side.
The champions had earlier dominated the match at the Bernabeu, but were frustrated by a new-look Valencia and some outstanding saves from goalkeeper Diego Alves.
"It's always important to start with a win," Vilanova, who has stepped up from assistant to replace Pep Guardiola, told a news conference.
"It has been a perfect night because David has returned with a goal. I am very pleased but this is only the beginning."
Although it is the first round of matches, Barca's two-point advantage over Real is significant because Jose Mourinho's side only dropped 14 points in their 38 games last term on their way to a league record tally of 100.
Earlier, Gonzalo Higuain gave Real a 10th-minute lead from Angel di Maria's cross and, in stifling temperatures, they comfortably held Valencia at bay until just before the break.
Keeper Alves was Valencia's outstanding player in the second half, keeping out a string of efforts as Real raised their game and pressed but their opponents held firm to help earn a point for their new coach Mauricio Pellegrino.
"I expected more from my team," Mourinho told a news conference. "We haven't played a great game, but we did enough to win."
The visitors were three goals up in 31 minutes before last season's Europa League and King's Cup finalists responded in the second half.
Bilbao's Oscar De Marcos got a goal back just after the break and Mikel San Jose then struck twice to level the scores with 14 minutes to go before Betis finally triumphed late on through Alejandro Pozuelo and Jorge Molina with his second.
