Barca’s 10th consecutive success in all competitions, their longest winning streak since the end of 2010, put them in a strong position to advance to May's final after the return leg at the Madrigal in three weeks.
They would meet Athletic Bilbao, whose 23 Cup triumphs are only bettered by Barca’s 26, or 2006 winners Espanyol, who drew 1-1 in their first leg at Bilbao's San Mames stadium.
Uruguay forward Suarez is taking time to settle in Spain after joining Barca from Liverpool in the close season but although he has yet to hit scoring form himself he has impressed with his link-up play and creativity.
He robbed the ball in the 41st minute at the Nou Camp before sprinting forward and laying a perfect pass into Messi’s path for the Argentina captain to sweep a first-time shot past keeper Sergio Asenjo.
It was harsh on a well-organised Villarreal side, who had lost captain and midfielder Bruno Soriano to what looked like a serious leg injury around 10 minutes earlier.
Barca squandered a series of chances before the visitors levelled three minutes into the second period after home keeper Marc Andre Ter Stegen let a long-range Manuel Trigueros shot straight through his hands.
Andres Iniesta put Barca back in front a minute later when he fired a loose ball low into the net and Gerard Pique made it 3-1 in the 64th minute with a header from a Messi corner.
Messi would normally have taken the spot-kick when Barca were awarded a penalty for a Mateo Musacchio handball in the 70th minute but after he stepped aside to allow Neymar a chance the Brazil captain had his effort well saved by Asenjo.
Suarez won more appreciation from the home support in stoppage time when he almost caught Asenjo off his line with a lobbed shot from near the halfway line but the ball drifted narrowly wide with the keeper beaten.
"It was a good win but it could have been better," Pique said in an interview with Spanish television broadcaster Canal Plus.
"We know the Madrigal is a tough stadium to play at and we still have to win the tie," added the Spain international. "We are playing with a lot of confidence at the moment and that means we can react quickly when we concede."
Bilbao took the lead in the 11th minute at the San Mames thanks to a neat finish from forward Aritz Aduriz before Espanyol midfielder Victor Sanchez smashed in an equaliser 10 minutes before the break.
(Writing by Iain Rogers, editing by Ed Osmond and Justin Palmer)
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