The Perth side were the dominant team early on but fell behind to Obika's overhead kick late in the first half.
And Mullen made sure of St Mirren's second victory of the season with his goal after 62 minutes, leaving St Johnstone still seeking their first win of the campaign.
Buddies captain Stephen McGinn made his first league appearance of the season, with Ryan Flynn dropping out of the squad due to injury. Tony Andreu replaced Ilkay Durmus in their only other change from the side that lost 2-0 to Motherwell.
The visitors also made two changes from their 2-2 draw with Ross County. In came Jason Holt and Drey Wright for Alistair McCann and David Wotherspoon, who both dropped to the bench.
It was St Johnstone who started the brighter of the two sides and a ferocious drive from Murray Davidson went only narrowly wide.
The Perth side continued to look dangerous in attack. Anthony Ralston saw his shot blocked after a promising move before Matty Kennedy dragged an effort wide of goal when he really ought to have scored.
Both teams then claimed half-heartedly for penalties. Kennedy felt Paul McGinn had stood on his toe as he looked to turn in the box, while Danny Mullen also appealed for a high boot in the face from Jason Kerr. However, referee Nick Walsh was not interested in either.
St Mirren made the breakthrough after 37 minutes. A hopeful free-kick into their box was not cleared and, after Kirk Broadfoot had headed the ball across goal, Obika pulled off the perfect acrobatic kick to put the home side ahead.
They could have had a second in the opening minute of the second half. Calum Waters slipped a pass in to Andreu whose drive was well stopped by Zander Clark.
The goalkeeper was helpless, though, when Mullen clipped a shot past him soon after and would have been grateful to see it strike his far post and bounce to safety.
But Mullen got his goal in the 62nd minute as St Mirren all but sealed the win. Broadfoot was again involved as he set McGinn clear down the right wing.
The full-back played in the perfect cross and Mullen smashed the ball past Clark with a first-time effort.