Forest were the better side for most of the evening in South Wales and Silva's third goal of the season was enough to earn them a vital three points which moves them up to third in the Sky Bet Championship before Wednesday's round of fixtures.
Sabri Lamouchi made one change to the side that drew with Queens Park Rangers on Saturday as Ryan Yates came into the team in place of Adama Diakhaby who dropped to the bench.
Forest had the majority of possession in the game's opening stages as Cardiff were happy to stand off them but it almost cost the home side the opening goal. Sammy Ameobi was given too much time and space to drive towards the edge of the penalty area but he drilled his shot inches wide of Alex Smithies' left-hand post.
The Reds certainly should have been in front just after the 20-minute mark. Joe Lolley pulled the ball back from the right-hand side to Lewis Grabban, who was waiting just behind the penalty spot, but the striker dragged his effort wide when he should have done much better.
Cardiff had a strong clam for a penalty on the stroke of half-time as Matty Cash appeared to trip Josh Murphy inside the box but referee Stephen Martin waved away the appeals as the first half ended goalless.
Lamouchi made a change at half-time as Alfa Semedo replaced Ryan Yates and within four minutes of the second half, Forest had the lead. Grabban delivered an excellent cross from the right-hand side and Silva was arriving right on time to sweep the ball beyond Smithies and into the net.
The home side had a good chance to level midway through the second half as Junior Hoilett got on the end of a terrific right-wing cross from Will Vaulks but the former QPR man lashed his shot well wide at the back post.
It could have been game, set and match for Forest with 13 minutes to play. Grabban worked tremendously hard to chase the ball down the left-hand channel and hold it up before playing a clever pass inside to Semedo. The Portuguese moved the ball smartly inside the box but his finish was blocked well by Smithies.
If they hadn't made the result safe before, Forest should have done so in the final minute of stoppage time. Grabban raced away down the right-hand side and drilled the ball across the face of goal but neither Lolley nor Tyler Walker could get onto the end of the cross to tap it into an empty net.
That miss wasn't to matter though as Forest recorded their first win at the Cardiff City Stadium since 2010 and achieved a crucial three points in the race for promotion.
Forest: Samba; Cash, Worrall, Figueiredo, Ribeiro; Watson (c), Yates (Semedo 46'); Lolley, Silva (Walker 90'), Ameobi (Diakhaby 81'), Grabban
Subs not used: Muric, Carvalho, Jenkinson, Dawson
Goal: Silva 50'
Bookings: Ameobi 70', Worrall 84'
Cardiff: Smithies; Sanderson (Bacuna 81'), Morrison (c), Nelson, Bennett; Pack (Ralls 71'), Vaulks; Hoilett, Murphy (Ward 71'), Paterson; Glatzel
Subs not used: Etheridge, Flint, Smith, Whyte
Booking: Bacuna 82'
Referee: Stephen Martin
Attendance: 21,273 (872 away)