Kanoute put Sevilla in front on 18 minutes after being sent clear by Jesus Navas.
Sevilla should have been more than one up at the break but were denied by Vicente Guaita on numerous occasions and the hosts were nearly made to pay for those wasted opportunities as first Piotr Trochowski and then Julien Escude were sent off.
Escude's dismissal also resulted in a penalty for Valencia which Ever Banega hit against the post before the visitors were also reduced to 10 men for Artiz Aduriz's apparent stamp on Emir Spahic.
Unai Emery made six changes to the Valencia side that had drawn an excellent match 2-2 with Barcelona in midweek and they were thankful to goalkeeper Guaita after an extremely poor performance in the first half.
Kanoute had a goal wrongly ruled out for offside before opening the scoring with a fine finish at the end of a flowing counter-attack.
Navas then set Diego Perotti clear but he was denied by Guaita before Kanoute's poor pass prevented Alvaro Negredo from having a clear run on goal.
Valencia came out a different side after the break and Javi Varas was forced into two quick saves from Tino Costa and Jonas.
Guaita was then called into action again to deny Perotti before referee Muniz Fernandez began to take centre stage as Trochowski was the first player sent off for a second bookable offence after fouling Costa.
Varas then made an incredible save to keep his side ahead from another Jonas effort on the hour mark, but Valencia were awarded a penalty seven minutes later for Escude's desperate lunge on Aduriz. The Frenchman was duly dismissed but Banega's fierce effort came back off the post.
The visitors were next to see red when Aduriz was given a very harsh straight red for a stamp on Spahic despite there being minimal contact on the Bosnian.
Valencia continued to push forward with the extra man and Varas had one final save to make from Roberto Soldado in stoppage time to ensure the three points.