The south coast club's Welsh international goalkeeper, whose gaffe in the last Premiership fixture allowed Brian Deane to convert and Leicester to leave with a point, gifted Fulham an early lead.
With little apparent danger, Jones somehow spilled a through ball for Steve Marlet to sidefoot into an empty net.
Rory Delap equalised for Gordon Strachan's side but without those mistakes, Southampton would be over the 40-point mark.
As it is, despite being beaten just once in 10 matches now, they are yet to guarantee their Premiership future.
A draw, in fact, leaves both sides still involved in the relegation equation. Southampton began proceedings in the ascendancy with visiting goalkeeper Edwin van der Sar forced to punch Anders Svensson's deep free-kick clear.
But FA Cup semi-finalists Fulham, without a league win in Southampton in 67 years, were gifted an early lead through van der Sar's opposite number Jones' mistake.
A neat link-up on the halfway line between Steve Finnan and Sylvain Legwinski saw the latter lift the ball over the hosts' backline for Marlet to chase.
There was no apparent danger, however, when Jones slid out to collect five yards ahead of the France international forward.
But the Welshman inexplicably fumbled and as the ball ricocheted into Marlet's shins he was left to steer into an empty net.
Jones did ensure Fulham did not double their advantage in the 10th minute, though, blocking Barry Hayles's left-foot shot after the striker had cut in from the right.
Having taken advantage of a gift, the London side offered one of their own in the 20th minute when Andy Melville, recalled in central defence, under hit a back pass.
Marian Pahars, Southampton's 13-goal top scorer, seized on the opportunity but his shot brushed van der Sar's spreadeagled body and went the wrong side of the post.
Within two minutes, though, Southampton were level through Delap, who was returning from suspension.
The Republic of Ireland international received a return ball from Dodd in his stride, controlling with his right and lashing in with his left foot from just inside the area.
Yet, that moment of quality was rare in an error-strewn contest.
In fact, Southampton were lucky not to go behind again five minutes after restoring parity when Paul Williams was caught in possession by Hayles 25 yards out only for the latter to run too wide in rounding Jones and allowing the chance to pass.
Then, when a Southampton move broke down just inside their own half, Steed Malbranque advanced to drag off-target from the edge of the area.
At the other end, a Pahars volley was held high above his head by van der Sar before Matt Oakley's 20-yard drive flashed wide.
Southampton had drawn their previous three matches.
Fulham, meanwhile, went into Easter on the back of the worst form shown in the top-flight all season.
In fact, only bottom club Leicester have matched their run of six consecutive defeats.
Both teams were more circumspect on the resumption and the pace of the game subsided.
The first goal attempt of the half came in the 55th minute when Legwinski saw a long-range effort blocked by Paul Williams.
Fulham's usual fluency was not allowed to prosper in a niggly second period.
But Sean Davis did go close when his one-two with Malbranque gave him a sight of goal just after the hour, his effort just too high.
And Marlet's pace caused increasing uncertainty in the home defence.
It was Southampton, however, who could have gained the lead with 11 minutes to go as substitutes Jo Tessem and Davies combined.
Tessem's well-weighted pass allowed Davies to advance into the area and his sidefooted attempt beat van der Sar but struck the foot of the post and Brett Ormerod sliced the rebound into the crowd.
At the other end, another of the match's late entrants Louis Saha grazed the outside of Jones' right-hand upright after being teed up by Rufus Brevett's clever header.