The Irishman opened the scoring after just nine minutes at the Alexandra Stadium, before then doubling the visitors' advantage seven minutes after half-time.
Midfielder Adam Forshaw made sure of the result shortly after the hour mark, and though Chuks Aneke reduced the arrears with a 70th-minute penalty, Brentford comfortably claimed the points.
The win extends Brentford's unbeaten league run to 18 matches, and is the perfect confidence boost for Mark Warburton's charges ahead of the top-of-the-table clash with Wolves next weekend.
Michael Jacobs scored in each half for Kenny Jackett's men, who have now won five on the spin in League One.
The Preston players all wore shirts with Finney's name in place of their own, while a minutes silence turned into applause before kick-off.
The hosts had fallen behind to Mathieu Baudry's opener on the stroke of half-time, but Joe Garner's penalty in the 64th minute earned a share of the spoils.
Stevenage's Francois Zoko had cancelled out Wes Thomas' first-half opener in the 86th minute, only for Lee Frecklington to win it for the home side two minutes later.
Elsewhere, Bristol City twice surrendered the lead at Ashton Gate in a 2-2 draw with Tranmere Rovers, and Ben Reeves' 92nd-minute goal secured a 2-1 win for MK Dons over Oldham Athletic, who had Genseric Kusunga sent off for a second bookable offence in the 66th minute.
