Pretty sure a winning strategy has to exist, either for the starting player or the opponent. There are a finite number of possible games, and there's no draw possible.
That said, I don't have a solution. Just some thoughts.
I'd probably start by looking at pieces whose placement is the most limited. E.g. if there is only one spot left on the board with five successive empty squares and the opponent has a long piece left, it's useful to block that spot to prevent them from playing that piece. Similarly, if we have a piece left which can only go in one spot on the board, we want to play that piece first in order to prevent the opponent from blocking the spot. By default I'd leave the 1x1 pieces last since they can be played on any board position, but there may be situations where it's useful to use them earlier (e.g. if we can block multiple opponent's pieces with a single 1x1).