Cronulla lifted in the final 10 minutes with winger Sosai Feki scoring his second try of the afternoon before Ennis iced the game for the Sharks.
Should Souths beat the Sydney Roosters on Friday, it would put the Sharks in a must-win situation for their clash with Manly two days later with the Rabbitohs boasting a superior points differential.
One bright note for the Eels was the NRL's leading tryscorer Semi Radradra crossing for his 22nd try of the season to move within Steve Ella's club record of 23 set in 1982.
Cronulla's win comes after a tumultuous couple of days for the club over an alleged incident involving skipper Paul Gallen.
The lock fronted media on Friday, denying a News Corp Australia report which alleged he swore at a young boy who wanted his autograph.
The Sharks released a statement before Gallen's media appearance, saying the club would investigate the incident.
The family at the centre of the controversy have stood by their claims that Gallen swore at the boy.
- AAP