The hosts' hopes of a first win in five Premier League games were dashed on a frustrating afternoon against the resolute visitors.
Senegal international Ba fired against an upright after skipper Fabricio Coloccini had earlier hit the other post and then saw Vorm react brilliantly to turn away his 42nd-minute volley as the Magpies threatened to run riot before the break.
Winger Jonas Gutierrez also came within inches of a superb opener with the visitors desperately trying to repel tide after tide of attacks.
But on an afternoon when the Magpies paid tribute to former midfielder Gary Speed, they could not mark the occasion with a victory in front of a crowd of 51,767.
The Magpies have now taken just two points from the past 15 on offer, although a fixture list which handed them clashes with Manchester City, Manchester United and Chelsea on consecutive weekends is a major factor in that.
Swansea arrived on Tyneside having collected 15 of their 17 points to date at the Liberty Stadium, and they showed few signs of adding to their paltry return on the road during the opening 45 minutes.
The afternoon had begun with a rousing tribute to former Wales boss Gary Speed with his family and close friends Alan Shearer, Steve Harper and John Carver there to witness it, and it was Newcastle who harnessed the emotion to rip into the visitors.
They dominated throughout and had they left the pitch at half-time four goals to the good, Swans manager Brendan Rodgers, who worked under opposite number Alan Pardew at Reading, could not have had too many complaints.
Coloccini, back to lead the side following injury, came close to opening the scoring after 19 minutes when he dived to head Yohan Cabaye's free-kick against the post.
In-form striker Ba was denied by the other post four minutes later after acrobatically volleying towards goal from a Leon Best knock-down.
Ba was looking the man most likely to break the deadlock, and he saw another effort blocked by Swansea captain Ashley Williams after turning smartly with 26 minutes gone.
As referee Lee Mason grew increasingly impatient with the visitors' rugged response to the deluge, the Welshmen found themselves pinned back deep inside their own half, and it seemed only a matter of time before Newcastle got their noses in front.
Ba must have thought he had given them the lead when he connected sweetly with a 42nd-minute volley from Gutierrez's header back across goal, only to see Vorm pull off a fine reaction save.
But there would have been little the goalkeeper could have done to keep out Gutierrez's raking right-foot piledriver had it not swerved just wide of the target at the last moment a minute before half-time.
The home side left the pitch to warm applause, but knowing they had to make the pressure tell after the break if they were to end their winless run.
They almost did just that within two minutes of the restart when full-back Danny Simpson floated a cross to the far post, where Ba volleyed over under pressure from defender Ashley Richards with his appeals for a penalty for a push in the back going unanswered.
Kemy Agustien, a half-time replacement for Mark Gower, hacked Coloccini's 54th-minute header away as the Magpies' maintained an unrelenting quest for the opening goal.
Former Magpies winger Wayne Routledge briefly gave the visitors breathing space when he drilled a low cross across the face of Tim Krul's goal, but with little support, the danger was minimal.
Vorm collected Ba's looping 67th-minute header with little difficulty after full-back Davide Santon had made the most of the space he was gifted down the left to send in a cross.
But with the opening goal remaining elusive, the visitors sensed an opportunity and went close 16 minutes from time when Scott Sinclair fired just wide of the far post after managing to get himself one-on-one with Simpson.
Having earlier introduced striker Shola Ameobi, Pardew threw on younger brother Sammy and fellow midfielder Haris Vuckic in a last-ditch attempt to snatch the points, but although the home side battered away at their opponents as time ran down, they could not find the necessary quality to claim the win.