The result has left Barnsley in 22nd position in the Championship table - two points off 21st-placed Bolton Wanderers with every team in the second tier now having played 44 matches.
Barnsley were a threat on the counter-attack, but it was Forest that continued to look the most likely, and the hosts made the breakthrough in the 27th minute when a 25-yard curling strike from Lee Tomlin found the back of the net.
Forest then doubled their advantage in the 36th minute when Ben Brereton latched on to a through-pass from Tomlin before producing a cool finish.
The visitors had the opportunity to strike back on the hour when Kieffer Moore found space on the edge of the area, but the forward fired wide of the post.
Cash then struck the post down the other end of the field 10 minutes later as Forest looked to make sure of all the points.
Aitor Karanka's side continued to search for a third in the latter stages, and it arrived in the 90th minute when Apostolos Vellios tucked home from close range after Walton had kept out an effort from Joe Lolley.
Barnsley's next Championship fixture is at home to Brentford this weekend before finishing the campaign with a trip to Derby County on May 6.
NOTTINGHAM FOREST: Kapino; Fuentes (Osborn 46'), Fox, Mancienne, Lichaj; Guedioura, Watson, Bridcutt; Tomlin (Lolley 80'), Brereton (Vellios 80'), Cash
BARNSLEY: Walton; Cavare, Jackson, Lindsay, Pinillos; Potts, Gardner, Moncur (Mahoney 46'); Hammill (Thiam 71'), Moore (Bradshaw 87'), McBurnie