The striker, who signed a new contract this week, netted his 13th goal of the season to consign Alan Johnston’s stuttering side to their third successive defeat.
Killie’s Josh Magennis fired narrowly wide in the early exchanges but his team-mate Ross Barbour had to produce a well-timed tackle to prevent the lively Stewart from giving the home side the lead.
Dundee ‘keeper Scott Bain produced an excellent reflex save to keep out Lee Ashcroft’s header, following an Alexei Eremenko corner, but when Sammy Clingan was deemed to have fouled Jim McAlister, Stewart made no mistake from the penalty spot in the 19th minute.
The impressive Eremenko continued to try and galvanise a response from the visiting side, delivering a stinging drive which was beaten away by Scott Bain before Iain Davidson went close with a right-foot effort for Dundee.
Gary Harkins fired against the crossbar from 18 yards as the home side looked to make the game safe and McAlister scooped another great chance over the top as Killie failed to find a response despite a brief late flurry.