After an uneventful first half the hosts took the lead through Jordi Alba’s 58th-minute side-footed finish.
Gavi tapped home Raphinha’s cross to double the lead after 71 minutes and the Brazil winger rounded off the scoring by poking in Alba’s cross.
Defeat leaves Sevilla 16th, just two points clear of the relegation zone.
One bit of bad news for Barcelona was captain Sergio Busquets going off with an ankle injury after just four minutes.
Barca host Manchester United in the first leg of their Europa League play-off tie on 16 February.
Earlier on Saturday, title rivals Real Madrid missed out on the chance to cut Barcelona’s lead at least temporarily to two points, losing 1-0 to Mallorca.