South Africa posted 97 for five in 7.5 overs after the game was initially a nine-over contest, before more rain meant it was reduced to the minimum of 30 deliveries for England's innings.
They were set a revised target of 69, but only Jos Buttler managed to get any momentum as he scored 25 from 11 deliveries before he was Jansen’s second wicket.
The tall fast bowler produced figures of 2-18 in his two overs as the home side reached 54 for five.
"You can't read too much into that, the field was very wet in patches and we didn't execute as well as we could have (with the ball)," England captain Harry Brook said.
Asked if he thought the conditions were playable, Brook was non-committal.
"The big dogs above me make the decisions," he said.
England had sent the visitors in to bat, and Donovan Ferreira finished unbeaten on 25 from 11 balls, with Aiden Markram (28 from 14 balls) and Dewald Brevis (23 from 10 balls) also making handy contributions.
Seamer Luke Wood took 2-22 in his two overs, while Sam Curran, on his return to the side, managed 1-11 in his only over.
"Happy to have got a pretty good score on the board and naturally the bowlers were always going to be under pressure, so to see them back up some good plans was very pleasing," Markram said.
"Marco has been out of cricket for a while, and to run in with that energy and set the tone was very important for us."
The second match in the three-game series will be played in Manchester on Friday.