Even a fabulous last minute Lionel Messi free-kick couldn't prevent Barca falling five points behind Madrid.
Even a fabulous last minute Lionel Messi free-kick couldn't prevent Barca falling five points behind Madrid, who also have a game in hand, after Nicola Sansone had put Villarreal in front just after half-time.
The draw capped a poor return to action in the new year for Barca after losing the first leg of their Copa del Rey last 16 tie at Athletic Bilbao on Thursday.
"We are aware that we are Barcelona and we are obliged to fight for all the titles," Suarez told Spanish TV station Movistar.
Barca host Bilbao in the return leg on Wednesday hoping to avoid a first Cup exit to anyone other than Madrid since 2008.
"What matters now is the Cup game on Wednesday," added Suarez.
"We have to go out from the first minute to win the game and qualify."
Suarez had an unusually quiet night in front of goal as most of Barca's opportunities fell the way of his strike partners Neymar and Messi.
"They have shown all season that they are very difficult to play against at home. We had our chances in the second-half, but we couldn't take them until Leo's goal," added Suarez.
"We've let two points get away, but we just have to keep working hard."
Barca boss Luis Enrique believed his side had done enough to win the game despite a familiar lack of cutting edge up front and fragility at the back this season.
"Without doubt I think we deserved to win the game," he said.
"We played well. We are not on the most positive run and you have to do more to win these type of games, but I have no criticism for my players.
"They played a complete game and we deserved the three points."
And with still over half the league season to go, Enrique insisted Barca still have time to chase down Madrid.
"There is still half the season to go and I am sure we will have the chance to fight for the league," he added.
"The only chance we will have to fight for the title, though, is by getting back to winning soon. To keep playing like this but with a different result."
Off the field, rumours emanating from Croatia claimed on Sunday that Barca midfielder Ivan Rakitic could be on his way to Manchester City after being left out of Enrique's squad.
However, Barca vice-president Jordi Mestre said the Croatian was "loved" at the Camp Nou and that negotiations were ongoing to extend his contract beyond 2019.
"We are negotiating Rakitic's renewal," Mestre told Movistar.
"We love Rakitic as a player and as a person. At the moment he has a contract and is a Barca player."