Ernesto Valverde's men looked to be in for a frustrating outing as they struggled to turn their domination into something meaningful, but eventually Lionel Messi clicked into gear and provided Suarez with the chance to give Barca the advantage.
The first half was a long one for Valencia, which restricted Barca's flow of chances but faced a seemingly endless barrage and only managed to see 21 per cent of the ball.
For all of Barca's time in possession, however, they crafted little of note and struggled to break through Valencia's packed defence.
Things were slightly more open after the interval, though Barca's effectiveness in front of goal showed few signs of improvement, forcing Valverde to introduce Philippe Coutinho from the bench.
And with 23 minutes to go, Barca finally found a way through thanks to the combination of Messi and Suarez, securing a slender lead to take to Mestalla next Friday.
In one sense, Barca could not have started the contest much better, as they dominated Valencia and attacked relentlessly, playing some wonderful football at times as they enjoyed 79 per cent possession in the first period.
Yet, due to a combination of Valencia's stubborn rear-guard and Barca's attackers not quite clicking, the chances just did not arrive.
Opportunities threatened to present themselves on a couple of occasions, but an uncharacteristically heavy Messi touch saw him accidentally juggle the ball out of play in the 22nd minute, and Jose Gaya covered well to prevent Suarez collecting a through ball soon after.
The closest they went to the breakthrough was in the 39th minute, as Andres Iniesta flashed the ball across the face of goal, but goalkeeper Jaume Domenech managed to divert it just behind the lurking Suarez with his fingertips and Barca finished the first half without a shot on target.
Presumably fresh from choice words courtesy of Valverde, it took Barca less than two minutes to craft the first chance of the second period, as Messi thrashed a left-footed effort into the side-netting from a tight angle.
The match's first shot on target followed soon after, though it came at the other end – Dani Parejo testing Jasper Cillessen from distance.
Coutinho replaced Aleix Vidal just before the hour mark, but unsurprisingly it was Messi who stepped things up to set up the equaliser, clipping a ball to the back post, where Suarez headed home.
Messi nearly added a second four minutes from time, curling a 25-yard effort just wide of the bottom-left corner, but Valencia never threatened to make them rue that miss as Barca cruised to victory.