A goal in each half from Franco Vasquez in the 36th minute and Luis Muriel in the 50th minute put Sevilla 2-0 up.
Suarez gave the Catalonia giants hope when he scored to make it 2-1 in the 88th minute.
And in the 89th minute, substitute Messi scored for Barcelona to secure a share of the spoils for his side.
Barca have now gone 30 league games without defeat this season, winning 23 and drawing seven games.
The draw took their points to 76, and they are now 12 points ahead of second-placed Atletico Madrid.
Atletico will hope to cut Barcelona’s lead to nine points when they host relegation threatened Deportivo La Coruna on Sunday.