Like Moonlight Sonata by Beethoven, or River Flows through You by Yiruma. Any suggestions?

How about Rachmaninoff prelude op. 32 no. 10?

hxxp: www. youtube. com / watch?v=IxFI2cphuN0

Its sad. . . yet beautiful

And by the way, "River Flows through you" by Yiruma is not classical.
And you should call them pieces, not songs. . . Just letting you know

Try these too:

Samuel Barber - Adagio for Strings
hxxp: www. youtube. com / watch?v=CA93ybVGCeg&feature=related
It was voted saddest classical piece ever once.

Schindler's List theme song
hxxp: www. youtube. com / watch?v=XLK5OWU2YGw&feature=related

Faure's Elegie for Cello and Piano

have you ever watched Secret That Cannot Be Told (movie)?
it has lots of good songs in them that are "sad".

or try preludes, by Bach. i don't know if it's depressing enough for you, but those are songs that are nice to listen to. a good website to download classical songs, is Classic Cat.
and sonatas, my sister used to play them a lot, and i loved them. every single one of them.
Fantasia and Fugues? beautiful!
Pachelbel's Canon' in D?
Marriage D' Armor?

and lots, lots more.

Pathetique Sonata mov. 3 by Beethoven maybe. . .

"Thais Meditation" by Massenet.
"Serenade" by Schubert
"Claire de Lune" by Debussy
"Bella's Lullaby" from Twilight
"Andante Cantabile for Cello and Strings orchestra" by Tchaikovsky
"Partita for Cello" by Bach

try to find some pieces for cello. . cello's sound which is very deep is very suitable for melancholy pieces. So it must be easier to find "sad songs" for cello. .

Saint-Saens - Requiem : Agnus
Saint-Saens - Samson et Dalila : Introduction.
Grieg - Peer Gynt suite : Abduction & Ingrid's Lament - Aase's Death.
Grieg - Heart Wounds Op. 34
Chopin - Polonaise in C minor Op 40 No. 2.
Chopin - Piano Concerto No. 1.
Camargo Guarnieri - Ponteio No 49
Beethoven - Piano Concerto No. 3.
Beethoven - Symphony No. 7, op. 92 - 2nd mov.
Brahms - Piano Concerto No. 1.
Rachmaninov - Piano Concerto No. 2.
Mozart - Piano Concerto No. 23 - 2nd mov.
Mozart - Synfonia Concertante K 364 - 2nd mov.
Tchaikovsky - Meditation de Souvenir d'un lieu cher.
Faure - Pavane, Op. 50