After Temba Bavuma had won the toss and opted to bowl, spinner Maharaj took 4-22 in 5.3 overs and seamer Mulder finished with figures of 3-33 in seven overs as the home team were skittled for a paltry 131 all out.
Only opener Jamie Smith made any score of significance with 54, but only three other players reached double-figures with Joe Root (14), captain Harry Brook (12) and Jos Buttler (15) failing to kick on after getting starts.
The tone was set early when left-arm quick Nandre Burger dismissed Ben Duckett in the third over while Lungi Ngidi took the wicket of the dangerous Root after a smart diving catch by wicketkeeper Ryan Rickelton, who held at the second attempt.
Maharaj came into the attack and continued his good form from the tour of Australia and initiated a collapse that saw the hosts losing their last seven wickets for just 49 runs.
Aiden Markram then gave England no respite in the run-chase, reaching a quick-fire fifty from just 23 balls with opening partner Rickelton at the other end content to enjoy the show.
The duo put on 121 with Markram dominant before falling for 86 from 55 balls while Temba Bavuma and Tristan Stubbs also lost their wickets, all three falling to spinner Adil Rashid.
Dewald Brevis walked to the crease and smashed his second delivery for a six to finish the contest in the 21st over with Rickelton unbeaten on 31.
The Proteas can wrap up a series victory with the second game set to take place at Lord's on Thursday.