'''96 Litespeed Catalyst, '05 Litespeed Firenze, '06 Litespeed Tuscany, '12 Surly Pacer, All are 3x8,9 or 10. It is hilly around here!
Yes it will work. A 10-speed chain is narrower than what you now use so it can be a bit reluctant to shift if you shift under a big load. Shimano used to recommend that you "shift sitting down", i.e. with relatively light pedal pressure, when using a 9 or 10-speed chain on an 8-speed or older crank.
If you change only the brifters and cassette, you might get by with a
9spd chain, cheaper and likely a bit more durable than a 10spd. CW
won't care, cassette +/- on that, but worth a try IMO. FWIW 8spd
chain works fine on 9spd IME.
A problem you can have with this setup is a 10 chain dropping between the large and small front chainrings as you shift. I had this problem going to a 10 DA chain when I was using a combination of 8 and 9 speed front rings. I ultimately had to go to 10s chainrings on the front - problem solved. FSA chainrings on the DA crank worked great, but of course weren't "stock."