Chippa were the brighter in the opening exchanges Chippa and Thabiso Lebitso had a shot towards the near post from a tight angle repelled by Swallows goalkeeper Thakasani Mbanjwa.
Swallows took the lead against the run of play in the 20th minute when Kagiso Malinga fired in at the near post from a tight angle.
Chippa equalised shortly after the half-hour mark as Lebitso tucked home after some neat interplay in and around the Swallows box.
The Birds regained the lead midway through the second half when Lloyd Kazapua misjudged the flight of the ball resulting in Malinga stealing in to complete his brace.
The Chilli Boys began to play with greater urgency in looking to level matters but after some pressure Etiosa Ighodaro headed narrowly over the top from a corner.
In the end, Swallows did enough to see the game out and claim their first away win of the season.