The bottom side's hopes of securing their first victory at the eighth attempt were dealt an early blow when County scored inside 10 minutes.
The Staggies salvaged a draw when Blair Spittal levelled with a spectacular free-kick, but Saints will feel they did enough to have claimed all three points.
Questions had to be asked of the visitors' defence when they failed to mark Joe Chalmers after Billy McKay's initial shot from Ross Stewart's cross from the right was pushed onto the post by Zander Clark.
Chalmers should have been closed down much quicker before being able to follow up from midway inside the 18-yard box to convert the rebound into a goal.
The visitors, clearly shaken, looked even more nervy than they had done in the opening exchanges and risked falling further behind.
Stevie May stepped up to take the resultant penalty kick but his shot lacked the necessary power to beat Ross Laidlaw when the keeper guessed correctly and dived to his left to smother the ball.
However, May atoned 14 minutes later when he scored the equaliser, finishing off Swanson's perfect delivery from the right from around seven yards for his first goal since returning from Aberdeen last month.
The goal had the effect of immediately lifting the Saints players' spirits and County suddenly found themselves to be the team under pressure.
May in particular became much livelier and began to display some nifty footwork as he attempted to carve out fresh openings for himself.
Saints were partially assisted by their rivals' apparent lack of energy and urgency which contrasted sharply with their early play.
But it was Kennedy's persistence rather than County's lethargy that enable Saints to edge in front after 51 minutes.
Kenny gathered the ball just inside the County half of the pitch and made a determined run before unleashing a shot from 25 yards that beat Laidlaw via the post.
But County hit back after 68 minutes when Spittal scored with a cracking free-kick from 25 yards that gave Clark no chance.
Saints dominated the final 20 minutes, when substitute Chris Kane just missed the target with a dipping effort and Kennedy forced a fine save from Laidlaw, and Kane fired another gilt-edged chance over the bar.