The weather intervened after one over of the post-lunch session and a number of showers prevented any resumption.
David Willey and Steven Patterson claimed a wicket each as Surrey were reduced to 36-2 after winning the toss.
Ansari (35*) and Finch (34*) collected 13 boundaries between them as they added 59 to take the score to 95-2.
Defending champions Yorkshire began the match in fifth place, 15 points behind leaders Middlesex, with Surrey next to bottom and 17 points adrift of seventh-placed Notts.
The visitors named an unusual XI, with only four specialist batsmen and five all-rounders, including Willey, who was playing his first Championship game since the reverse fixture at Headingley in early May.
He made the initial breakthrough by trapping Rory Burns lbw in the eighth over and wicketkeeper Andrew Hodd then took a superb diving catch to remove Dominic Sibley.
Finch, playing against his former county, marked his arrival with some thoroughbred strokes, and Ansari lost little by comparison as they steadied the innings before the weather closed in and play was abandoned for the day at 17:00 BST.
Zafar Ansari and Aaron Finch shared a half-century stand for Surrey before rain washed out play on the opening day against Yorkshire at The Oval.