They did, and they did well in the 7th too, which must be what I remembered. I had often thought that the Sox "made their hay" when a team's starters were in the 3rd time through the order and then when the relievers came in. The top 4 innings were the 5th, 7th, 1st, & 6th.
I'd always heard that more runs were scored in the 1st than any other inning because 1) the pitchers weren't settled in, and 2) that's the only inning in which a team can say in advance who's coming to bat.
This is from 2011 so I don't know if it still carries or not....
https://www.beyondtheboxscore.com/2011/7/3/2255959/all-innings-are-not-created-equal-how-run-scoring-varies-by-inning
Thanks Bell (and Moon). I need to explore more. I had no idea all those stats were compiled and available.