Manager Vilanova, back at the helm after surgery on his parotid gland after contracting cancer for the second time, was given the easiest of nights as Barca coasted to a one-sided 4-0 win.
They had scored all of their goals by the half-hour and after that it was procession football, with Pedro twice denied a hat-trick by the offside flag while other chances also passed.
After Xavi had opened the scoring, Pedro then chimed in with his brace before Lionel Messi added a penalty.
The win leaves Barca 11 points clear at the summit after nearest challengers Atletico Madrid could only draw 1-1 at Real Mallorca.
Raul Garcia broke the deadlock in the 72nd minute for an Atletico side shorn of star striker Radamel Falcao due to suspension, and Mallorca punished the visitors' inability to see them off as Kevin Garcia popped up with an equaliser three minutes from time.
Third-placed Real Madrid stay 16 points behind Barca after defeating Real Sociedad 4-3 despite playing for 84 minutes with 10 men.
Karim Benzema had swept the champions into the lead after 90 seconds but Jose Mourinho's decision to once again leave goalkeeper Iker Casillas on the bench backfired six minutes in as replacement Antonio Adan brought down Carlos Vela inside the area and was sent off.
Xabi Prieto slotted home the penalty, and both sides exchanged further goals before half-time as Sami Khedira put Madrid back in front only for Prieto to level five minutes later.
Cristiano Ronaldo looked to have sealed the points with two goals in two minutes midway through the second half, but Sociedad were given another lifeline when Prieto completed his hat-trick with 15 minutes remaining.
The visitors themselves were reduced to 10 men on 80 minutes as Dani Estrada was shown a second yellow card for bringing down Ronaldo and the Basques failed to muster another effort on goal.
In the day's early kick-off, two goals from Iago Aspas and a wonderful strike from Alex Lopez were enough to hand Celta Vigo a 3-1 win over Valladolid.
Aspas enhanced his burgeoning reputation with a fine run and finish after eight minutes, but Valladolid were level moments later when Alberto Bueno converted from the spot after Javi Varas had brought down Javi Guerra.
Aspas restored the lead with a penalty of his own on the half-hour and Lopez sealed the points seven minutes into the second half with a brilliant strike from 25 yards.
Friday & Saturday's Round-Ups
On Friday, Real Betis did their chances of securing Champions League football next season a world of good with a hard fought 2-1 win over Real Zaragoza.
A day later, Deportivo La Coruna gave new manager Domingos a winning start on Saturday night as the struggling team beat high-flying Malaga to move closer to an escape from the La Liga relegation zone.
A first win since early November for the Galicians was secured by a goal from Pizzi 12 minutes into the second half, when he turned Martin Demichelis before sending a shot high past goalkeeper Willy Caballero.
Deportivo remain in the relegation zone only on goal difference, while Malaga stay in fourth place, level on points with Real Betis who beat Real Zaragoza last night.
Levante are in pursuit of Malaga and are just one point behind now after coming from behind to beat Athletic Bilbao 3-1 at the Ciutat de Valencia.
Athletic's Aritz Aduriz headed in Andoni Iraola's cross for his 15th goal of the season early on, but Levante levelled with a thunderous strike from Christian Lell.
Athletic were reduced to 10 men after Aymeric Laporte fouled Roger to deny him a clear run on goal, and Levante made their man advantage count almost immediately as Vicente Iborra scored to give the hosts a 2-1 interval lead. Levante made the points safe in the 70th minute when Nabil El-Zhar drilled in a low drive.
Valencia began the year positively as second-half goals from Jonas and Pablo Piatti helped them come from behind to beat Granada 2-1.
Granada had Mikel Rico sent off shortly after the half-hour mark but went ahead four minutes into the second half when Brayan Angulo volleyed in.
Jonas headed an equaliser from a corner delivered by Piatti, who then redirected Roberto Soldado's cross-shot into an empty net eight minutes from time.
Valencia climbed to seventh place, with Granada still just a point above the relegation zone.
Sevilla eased their relegation concerns with a 1-0 win over bottom-of-the-table Osasuna in the final match of the day.
Defender Emir Spahic scored the only goal, after 83 minutes, to haul Sevilla seven points clear of the bottom three.
Friday's La Liga Result
Saturday's La Liga Results
Sunday's La Liga Results
Monday's La Liga Fixture
Rayo Vallecano vs Getafe