The home team totalled 331 in their second innings before Pakistan, chasing 191 to win, reached the close on eight for nought.
Asif bowled with pace and accuracy to take five for 76 from 38 overs.
"We have a good chance and we hope we're going to win although it's difficult to say which team has the upper hand," Asif told reporters.
South Africa lost five wickets for 95 runs before fighting back with a last-wicket stand of 41 by Andre Nel and Makhaya Ntini. Jacques Kallis top-scored with 91.
The hosts resumed on 115 for three and Kallis and Ashwell Prince took their fourth-wicket partnership to 56 before Prince was trapped lbw for 22 by leg spinner Danish Kaneria.
Herschelle Gibbs helped Kallis add 78 for the fifth wicket but Gibbs (40) edged Asif to Younis Khan at second slip after lunch.
KALLIS DROPPED
Kallis was given a life by wicketkeeper Kamran Akmal on 63 after edging a delivery from Kaneria.
The circumspect Kallis was eventually trapped lbw by Asif seven overs after lunch.
Mark Boucher and Shaun Pollock raised the scoring rate in a stand of 80 for the seventh wicket which ended when Pollock was caught and bowled by Kaneria for 36.
Boucher hit 46 before edging a catch to Younis at slip to earn Asif his fifth wicket.
Paul Harris fell for nought when he turned Kaneria into the hands of substitute fielder Faisal Iqbal at short leg.
Nel and Ntini then frustrated Pakistan before Ntini (18) lofted Kaneria to Yasir Hameed at point.
Nel finished on a career-best 23 not out.
Kallis said: "The wicket did something all day, the ball was nipping around.
"We're happy with 191, we think we can defend that. There is a bit in it for the bowlers and I think it's going to be a good contest."
Pakistan were without fast bowler Shoaib Akhtar because of a hamstring injury.
Shoaib took four for 36 in the first innings in his first Test appearance for 11 months.
