The 29-year-old stand-off featured in every match for Frank Hadden's team during their campaign which came to an end with Sunday's 19-13 defeat to Argentina in the quarter-finals.
Dan started four of Scotland's five matches and came on as a replacement in the defeat to New Zealand at Murrayfield, when Hadden selected a weakened team to protect his players for their crunch clash with Italy.
Dan was among Scotland's try-scorers in their 56-10 pool victory over Portugal and kicked six goals from seven attempts during the tournament.
The Glasgow Warriors man finished ahead of Rory Lamont and Simon Taylor who were second and third in the voting.
Scotland polled their squad after each match, with the player of the tournament outcome based on the aggregate result.