The forward's first league goal of the campaign also ensured that Espanyol's third coach of the season, Abelardo Fernandez got off to a positive start as the club seeks to avoid relegation
Espanyol looked to stifle Barca on the ball, but although the league leaders had plenty of possession they did little to threaten Espanyol before David Lopez put the home side ahead with a flicked header following a free kick from the right.
Barca coach Ernesto Valverde looked to inject some more urgency into his side, replacing Ivan Rakitic with Arturo Vidal and he got his reward in the 50th minute when Suarez, who had earlier hit the post, produced a sharp finish at the near post.
The Uruguayan then produced the cross that allowed Vidal to head Barca in front in the 59th minute.
Espanyol keeper Diego Lopez twice saved well to deny Suarez a second goal of the night, before Frenkie de Jong was sent off with 15 minutes remaining after receiving his second booking of the game.
Wu Lei came on as a substitute for Espanyol a minute before De Jong's sending off and quickly tested Neto with a header which the keeper saved well
However, Neto could do nothing in the 88th minute when Wu Lei found space on the right of the Barca area and drilled a low shot into the corner of the net to spark wild celebrations in the stands.
Real Madrid had defender Raphael Varane and goalkeeper Thibaut Courtois to thank as they won 3-0 away to Gefate.
Varene headed in Madrid's two opening goals - the first with the collaboration of Getafe keeper, David Soria, while Courtois kept Getafe at bay with a series of impressive saves, before Luka Modric scored Madrid's third goal deep into injury time.
All of the goals came in the first 18 minutes as Atletico Madrid beat Levante 2-1 in the Wanda Metropolitano stadium.
Angel Correa opened the scoring in the 13th minute, only for Roger Marti to equalize for the visitors three minutes later with his ninth goal of the season.
Felipe Monteiro then netted the decisive goal two minutes later and Atletico were content to keep Levante at bay for the rest of the game, although the visitors almost snatched a point in the last seconds only for Jan Oblak to produce a stunning save to deny Enis Bardhi's header.
Maxi Gomez's 27th minute header gave Valencia three points at home to a hard-working Eibar and the visitors' day was made worse in injury time when defender Paulo Oliveira was sent off after seeing a second yellow card.
Friday saw two games with Sevilla held to a 1-1 draw at home to Athletic Club Bilbao, who took the lead thanks to a first half goal from Ander Capa, only for Unai Nunez to put the ball into his own net while trying to cut out a dangerous cross on the hour.
Meanwhile Leganes' improvement continued under Javier Aguirre as the side from the south of Madrid took a 2-2 draw from their visit to Valladolid with Martin Braithwaite and Roque Mesa's goals for the visitors cancelled out by two goals from Enis Unal.