It is Arrows' second win of the season and moves them up to 11th on the Premiership table.
The first chance of the match fell to the hosts in the 13th minute, but Daouda Kassaly was able to hold on to Nkosinathi Ogle's weakly struck shot.
Two minutes later Petrus Ngebo managed to get behind the Arrows defence on the left, but his finishing left him down and he volleyed his shot over the crossbar.
Justine Zulu was the next man to hit the target in the 27th minute, after pushing forward from the middle of the park, but his shot went straight into the goalkeeper's gloves.
Three minutes later, Zulu released Ogle on the right, who then shifted inside before firing a low shot at goal, producing a comfortable save from the Niger international.
Smith made it 1-0 just after the half-hour mark, showing composure inside the box before firing into the roof of the net.
Chippa came close to finding the equaliser two minutes later, when Diego Brown delivered an inswinging free-kick towards Ngebo, but his header was deflected wide.
Arrows continued to be the better side in the second half and came close to doubling their lead in the 48th minute, but Ogle's free-kick was deflected narrowly over the bar.
Chippa's best chance of the match fell for Welcome Qalanto in the 50th minute, after he collected a pass from Cole Alexander, but his shot was parried by Pavels Steinbors.
Arrows took advantage of that missed opportunity two minutes later when Smith tucked a low shot past Kassaly, after receiving a pass from Ogle.
The hosts finished as the stronger side, with Ogle forcing a save from Kassaly in the 65th minute, while Philani Shange was also denied by the Niger international five minutes before full-time.