Suarez opened the scoring in acrobatic fashion in the 27th minute and goals from Arturo Vidal, Ousmane Dembele and Lionel Messi – his first of the season from a trademark free-kick – sealed a comfortable victory.
The win was Barca's fourth in a row in all competitions since a shock 2-0 loss at Granada on September 21 and lifts Ernesto Valverde's side to second place in LaLiga, two points behind leaders Real Madrid.
However, the win was marred by the late dismissals of Dembele and Ronald Araujo, the former receiving a second yellow card for dissent after last man Araujo had been sent off for hauling down Javier Hernandez.
Sevilla began the game brightly and were left to rue a number of missed chances, with striker Luuk De Jong the main culprit.
De Jong saw his first-time shot superbly saved at his near post by Marc-Andre Ter Stegen after 11 minutes but should have broken the deadlock seven minutes later when he fired wide from six yards with the goal at his mercy.
Messi then curled a free-kick over the crossbar before De Jong squandered another good chance, heading a cross from Lucas Ocampos into the ground and over the bar from close range.
Moments later Suarez made the visitors pay the price for their profligacy, the Uruguay international connecting perfectly with an overhead kick to dispatch Nelson Semedo's cross into the bottom corner.
Vidal doubled the lead five minutes later when he slid in to convert a pin-point pass from Arthur as Sevilla players appealed in vain for offside and Dembele soon made it 3-0 with a delightful right-footed finish from 12 yards out.
Sevilla's miserable evening in front of goal continued as Nolito headed a cross from Jesus Navas tamely at Ter Stegen before half-time and the interval did nothing to change matters.
Five minutes into the second half, De Jong dispossessed Arthur on the edge of the penalty area but could only hit the post with a right-footed shot and it was no surprise when the 29-year-old was substituted with 25 minutes remaining.
After two previous attempts Messi finally found his range 12 minutes from time and curled a superb free-kick beyond Tomas Vaclik to open his account for the season.
That should have been the signal for the game to peter out but Araujo brought down fellow substitute Hernandez on the edge of the area as he attempted to run through on goal and was sent off.
Dembele, who had been booked early in the second half, was quick to remonstrate with the referee but appeared to overstep the mark and was shown a second yellow card.