Replying to SA's 203, the Bulls were all out for 238 before Bartlett crashed through the home side's top order on Tuesday's second day of play at Adelaide Oval.
Bartlett also took a smart slips catch as the hosts slumped to 4-22 before wobbling to 7-104 at stumps, a lead of only 69 runs.
The Bulls paceman bowled Henry Hunt (two) with a sublime delivery and, two balls later, dismissed Jake Weatherald (four) when the Test discard flashed outside off stump and was caught at gully.
In Cameron Gannon's next over, Bartlett collected a slick, low catch at first slip as Nathan McSweeney fell for one. The SA skipper was clearly unhappy with the decision, with replays suggesting the ball hit his right leg.
Three overs later, Bartlett struck again when Jake Lehmann (eight) was trapped lbw.
The plight of the defending Shield champions worsened six overs later when Jem Ryan collected two wickets in two overs.
Ryan (2-22) bowled Jason Sangha (12) with an off-cutter and bowled Harry Neilsen for a duck when the lefthander attempted a leave but edged onto his stumps.
Liam Scott (33) and Ben Manenti (28no) put on 47 runs for the seventh wicket, with Bartlett ending the day with 3-33 from 10 overs and Gannon following his five-wicket haul in the first innings with 2-29.
Earlier, the Bulls established a first innings lead with Jack Clayton top-scoring with 66.
Clayton struck nine fours in his 129-ball knock before the left-hander was trapped lbw by Brendan Doggett.
A raft of Clayton's teammates failed to cash in after promising starts with Hugo Burdon (39), Jimmy Peirson (27) and Hayden Kerr (26) appearing well set before being dismissed.
Tailenders Mitch Swepson (19) and Cameron Gannon (17) added some handy late runs against an SA attack led by two-Test paceman Doggett.
The former Queenslander picked up 3-46 from 20 overs, Nathan McAndrew claimed 2-76 and part-time offspinner Sangha took 2-3, capturing the last two wickets.
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