I would suggest using a drip edge like Joe posted. You should be able to slide it under the shingles with out removing the gutter. If you can't hold the shingle and starter shingle up at the same time to nail it, try using a drill with a magnetic tip and some short deck screws. Be careful that the shingles are not brittle and crack. You could put some dabs of roof cement under the shingles to hold the drip edge if they won't bend without cracking.
May not be the approved installation method, but it will save making a bad situation worse.