Johnson reached the 1,000-point milestone with a free throw with 15:58 remaining in the second half. She becomes the 19th player in school history to record 1,000-career points.
Junior Darriel Gaynor connected on three three-point baskets to finish with nine points, two shy of her career high.
It was a back-and-forth contest that included five ties and 15 lead changes. In the opening half, MSU built a four-point lead with 8:46 to go on a Gaynor three.
LSU cut the advantage to 19-17 before State answered with jumpers by Johnson and Alwal to regain the four-point edge.
The Lady Tigers responded with an 11-0 run to go up 28-21 with 1:26 to go, but State closed out the last four points of the half to trail by only three at the intermission.
Jeanne Kenney opened the second period with a basket to put LSU up five, but State responded with a 7-0 spurt that Johnson capped with her milestone-setting free throws.
Neither team would lead by more than two until LSU claimed the lead for good with a pair of free throws by LaSondra Barrett that put the Lady Tigers in front 46-42.
Barrett scored the last 11 points for LSU to finish with 17 points.
MSU shot 36 percent for the contest, including 41.7 percent from three, with Alwal adding eight to go along with the 15 from Johnson and Gaynor's nine. Katia May, Porsha Porter and Shamia Robinson scored four each, while Catina Bett and Kendra Grant added two points each. Danielle Record added a free throw to cap MSU's scoring.
The Lady Bulldogs return home Sunday to face in-state rival Ole Miss at 2:30 p.m. The contest, which will be broadcast by SportSouth, will be the team's annual Think Pink Game presented by OCH.