I thought it would be enough. "I strongly believe that our servicemen will fulfill their mission with dignity and high professionalism also utilizing the extensive experience they have garnered in the last decade in Kosovo, Iraq and Afghanistan."
'We use the Present Perfect to show that something started in the past and has continued up until now'── quoted from http://www.englishpage.com/verbpage/presentperfect.html
So I think the present perfect should be used in your sentence.
If the sentence had begun 'Our servicemen fulfilled their mission...' it would have continued '...utilising the experience they garnered (or 'had garnered')...
I feel there's something wrong with the '[...] also utilizing [...]' part.
I think it should be something like 'I strongly believe that our servicemen will fulfill their mission with dignity and high professionalism, and also by utilizing the extensive experience they have garnered in the last decade in Kosovo, Iraq and Afghanistan.'
Is the original sentence fine as it is?
Thank you.