Reports earlier in the day suggested owners New England Sports Ventures were actively looking for a replacement for the 63-year-old, who has apologised to fans after appearing to question the commitment of the Anfield support.
When Kevin Davies headed in Matt Taylor's 43rd-minute free-kick things were looking bleak for Hodgson but a revitalised Torres volleyed home before England Cole tapped in his first league goal for the club he joined from Chelsea in the summer.
Meanwhile, Gareth Bale's lightning reflexes proved enough to extend Tottenham's perfect Christmas and send them back into the top four in as they won 1-0 at White Hart Lane in the London derby with Fulham.
The in-form Welsh winger reacted brilliantly to flick Rafael van der Vaart's free-kick strike past a wrong-footed Mark Schwarzer three minutes before half-time. Spurs missed plenty of chances to add to their lead but Michael Dawson also had to clear off his own line.
Otherwise, Fulham's lack of firepower once again cost them dear as they slipped to second bottom of the table, piling the pressure on manager Mark Hughes.
A Kenwyne Jones header and a freak Phil Jagielka own goal handed Stoke a 2-0 victory in a full-blooded clash against Everton.
Jones rose to power the ball past Tim Howard midway through the first half before Jagielka put through his own net for second time in successive appearances with 21 minutes remaining at the Britannia Stadium.
Darren Bent ended his personal goal drought as Sunderland bounced back from their horror show against Blackpool to defeat Blackburn 3-0.
The England striker glanced home a header eight minutes after Danny Welbeck's deflected 11th-minute shot had put the Black Cats in front to claim his 10th goal of the season and his first in six games. Substitute Asamoah Gyan sealed a victory which took Sunderland to 30 points for the season with a third at the death.
Freddie Sears put the seal on a sweet 2-0 victory for West Ham as they climbed out of the relegation zone and sent Wolves crashing to the foot of the table.
Winger Sears, a product of the West Ham academy and lifelong Hammers fan, struck in the 79th minute for his first goal since netting on his debut in March 2008.
West Ham were already ahead, but they owed a debt of gratitude to Wolves right-back Ronald Zubar for their 51st-minute opener.
Zubar let the ball trickle in off his knee after Carlton Cole had taken an air shot.