Sonko put Walsall in control with a 22nd-minute opener, but then had to come to the rescue of Richard Money's side with an equaliser to cap a frantic spell of three goals in three minutes.
Swindon, who had new boss Maurice Malpas watching from the stands before he formally takes charge on Monday, turned the game around midway through the second half.
Andy Nicholas' first goal of the season in the 65th minute fired them back on to level terms as he unleashed a fierce far post volley after Walsall had failed to clear a corner from the impressive Sofiane Zaaboub.
Walsall were still reeling from that setback when Swindon, who are now themselves unbeaten in six games, took the lead two minutes later.
Blair Sturrock whipped in a left-wing cross and the unmarked Craig Easton capped a driving display at the heart of midfield with a flashing header from six yards.
But their hard-fought advantage proved to be short-lived as Sonko needed just 60 seconds to draw Walsall back on level terms.
He made no mistake with a close-range header after being picked out by a great pin-point right-wing centre from Mark Bradley.
Walsall could then have gone on to win the game only for substitute Ishmel Demontagnac to be denied by a stunning 85th-minute save by Petr Brezovan.
Swindon had chances to break the deadlock themselves before Sonko gave Walsall the advantage.
Jon-Paul McGovern fired straight at Clayton Ince in the eighth minute. Ince then denied Sturrock four minutes later - diving low to his left to hold on to a low 20-yard shot - before the former Coventry City goalkeeper frustrated Zaaboub in the 20th minute.
Having ridden their luck, Walsall stunned Swindon when Sonko struck for the second time this season. Swindon failed to cut out a left-wing cross from Troy Deeney and his centre found its way to Bradley.
Brezovan did well to turn away Bradley's shot but there was no way he could recover in time to prevent Sonko stabbing home from six yards.
Walsall were fortunate to take the lead into half-time as Lee Peacock wasted a glorious chance to equalise in the 33rd minute.
After being picked out by a neat pass from McGovern, Peacock took the ball around Ince but, from an acute angle, he could only hit the post when faced with an empty net.