The visitors, who boasted the second best away defensive record in Europe before this encounter, looked set to hold on until the interval, but with the last kick of the first half Messi's deflected strike put the hosts back in front.
After the break, Barca toiled to find that crucial third, which eventually came via the boot of Suarez in the 69th minute, before Messi made sure of the victory with a Panenka penalty eight minutes from time.
The result sees Barca open a three-point lead over Real Madrid at the top of the table, with Real playing later against Granada.
Andres Iniesta returned to the Barcelona side after missing the last two matches with a groin injury, but even he couldn't unlock a well-organised Villarreal unit early on.
It took until the 21st minute for Barcelona to create a meaningful chance, but it was a telling one, as Neymar broke the offside trap, latching onto Messi's pass, before taking a touch and firing into the bottom corner.
The visitors responded well, and 10 minutes later they were level. Barca were playing a dangerously high line, and Roberto Soldado took advantage, threading a perfect pass into the path of Bakambu, who raced clear of Gerard Pique before calmly slotting home.
Villarreal should have taken the lead five minutes before half-time, but former Tottenham striker Soldado headed Roberto Soriano's cross wide with the goal at his mercy.
And Soldado was made to pay for his profligacy, as Messi stepped up to score a timely goal for Barca, after making space for the shot and slotting the ball into the bottom corner from just outside the box. The slight deflection on the effort wrong-footed Villarreal stopper Andres Fernandez.
Neymar then created an opportunity for himself just after the break, but was denied from the angle by a smart Fernandez stop.
Messi was also denied by Fernandez from the edge of the box, before Suarez finally broke Villarreal's second-half resistance, dropping the shoulder and firing into the bottom corner after a swift Barca break.
Barca turned on the style in the latter stages, with Neymar the main protagonist. He was clean through in the 77th minute, but inexplicably elected to chip for goal, which Fernandez dealt with easily.
The next bit of skill from the Brazil superstar created the chance that led to Jaume Costa being penalised for handball in the box as he tried to block a Suarez strike. Right on the byline, the ability to flick the ball over the defender's head just cannot be taught.
Not to be outdone, Messi chipped the penalty right down the middle to complete the rout and his record-breaking evening.