R-U claimed the early lead, finishing the first quarter three points ahead of RSM, at 9-6.
The Hawks answered back with a 10-point second quarter, while the Rockets scored five, making it 16-14 in RSM's favor at halftime.
With that, the Hawks took a 10-point lead into the final stanza, with the score 29-19.
"We played probably our best half of the year in that second half," RSM girls coach Nick Staab said. "We switched our offense a little bit and started driving to the hole, and we shot 17-of-18 from the line in the second half."
Free throws were the Hawks trump card -- they scored more than half their points from the charity stripe, going 25-of-30 overall, 83 percent.
RSM was just getting started, piling on the points. The score was 47-22 with two minutes to go.
"We just got a lot more aggressive in the second half. We started actually looking like basketball players," Staab said. "The first half we were kind of stagnant. We had a lot of nice looks at the basket, but we weren't hitting shots."
R-U added 10 more points in those final two minutes -- including nailing a half-court shot at the buzzer -- but it wasn't enough to close in on the Hawks, wo scored 20 in the final stanza.
"We played a pretty good half of basketball, the first half," R-U coach Mike West said. "Unfortunately you have to play two halves to win a game. The difference was they went to the line 30 times and they were 25-of-30. I believe we were 4-of-15."
RSM junior Lyndsie Schorg led in scoring and hit all her free throws for the night. She was 14-of-14 from the line, with 12 of those in the second half.
Free throws have been a focus of Hawk practice since Christmas break, RSM coach Staab said .
Senior Miranda Staab added another nine points for the Hawks. Staab and Schorg led in rebounds with nine.
Sophomore Alexis Bunkers grabbed six boards.
Schorg had six steals, senior Paige Harpenau had four, and seniors Anyssa Trejo and Makenzie Ricke each had three steals.
For Remsen-Union, senior Brittany Weikel led in scoring with nine points for the Rockets.
RSM's record improves to 6-6 overall and 3-3 in the War Eagle Conference.
"We told the girls we have a chance to win out if we play well to the end of the year," Staab said. "We've got to start building momentum for the tournament."
The Hawks don't play again until they meet up with Marcus-Meriden-Cleghorn at home Friday.
R-U's record moves to 5-8 overall heading into a game tonight (Tuesday) at Akron-Westfield.