Multan now has two wins from three games and is level at the top of the points table with title defender Quetta Gladiators and two-time champion Islamabad United.
The 2017 champion Peshawar, which was dismissed for its lowest PSL total of 123 in 18.3 overs, lost its second game and has two points from three matches.
Left-handers Rossouw and Khushdil Shah (43 not out off 29 balls) delighted a capacity crowd of 25,000 at the Multan Cricket Stadium with a match-winning unbeaten 77-run fifth-wicket stand as the home team raced to 124-4 off 14.5 overs.
"We executed our bowling plans very well and restricted the opponents," Multan captain Shan Masood said. "The intensity was right there although we dropped a catch off the very first ball, but both our seamers and spinners did a very good job."
After putting up a disappointing show with the bat, the Darren Sammy-led Peshawar struggled with its fielding as well. Sammy dropped James Vince in the first slip off the very first ball.
Rossouw also escaped twice when Rahat Ali dropped a low return catch off his own bowling and Liam Livingstone couldn't hold on to a tough chance in the deep when the batsman reached 46.
"They (Multan) used the conditions to their advantage and now we have to bounce back," Sammy said.
Earlier, after being put into bat, Tom Banton (4) couldn't capitalize after being dropped in the slip off Mohammad Irfan's first ball as veteran Shahid Afridi took a smart catch at short extra cover to dismiss the Englishman in the same over.
Pakistan Under-19 batsman Haider Ali made an aggressive 47 off 27 balls with six fours and two sixes, but once he holed out to Rossouw at deep mid-wicket off Tanvir in the 11th over, Peshawar's innings never got back on track.
Sammy (2) took 12 balls to get off the mark as both spinners Imran Tahir of South Africa and Afridi tied down the West Indian before the Peshawar captain miscued a pull and was caught at mid on.
Tanvir returned for his second spell and claimed the last two wickets as Peshawar finished on a below-par total.
Multan, which plays three matches at home, next faces Karachi Kings on Friday while Peshawar takes on bottom-placed Lahore Qalandars in Rawalpindi the same day.