Finn was only drafted into the squad last week, but underlined his credentials against South Africa A by taking four wickets for 34 runs at the City Oval.
England bowled out their opponents for 136 runs and scored 64 without reply before bad light curtailed the game and stumps were drawn.
Finn, who was forced out of the test series against Pakistan in October after injuring his left foot, was flown over to South Africa after impressing for England Lions in two comeback appearances against Pakistan A in Dubai.
His showing on Sunday will have increased his chances of joining James Anderson and Stuart Broad in a three-pronged seam attack for the first test in Durban.
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Ben Stokes, who is also making his way back from injury, took three wickets.
England will resume on Monday with Alastair Cook unbeaten on 36 and Alex Hales on 28.
(Reporting by Mark Gleeson; Editing by Toby Davis)
