Atwood accounted for three of Sam Houston's four RBIs. Burchett earned his ninth save of the season after shutting down Prairie View in the ninth after the Panthers loaded the bases with no outs.
Jackson (9-4) spread eight hits over eight innings, holding the Panthers scoreless for five innings after Prairie View took an early lead.
"This wasn't our best day offensively, but we'll take the victory," SHSU head coach David Pierce said. "We've been forcing it at the plate some, and that's not surprising for a young team. We got a strong performance from Justin for eight innings. He is a battler. And then Michael came in and shut them down that last inning. It was more exciting than I'd like, but that's how it always is in the regional."
The Panthers out-hit Sam Houston 9-7.
"Prairie View was putting the ball in play early in the count, but our defense did a great job behind me," Jackson said.
Held scoreless for the first four innings, Sam Houston took advantage of two Panther errors to take a 3-1 lead in the top of the fifth inning.
Catcher John Hale led off reaching second on an infield error. Second baseman Jessie Plumlee singled and shortstop Corey Toup's sacrifice bunt attempt was booted to load the bases. Hale and Plumlee scored on a singled by center fielder Colt Atwood. Toups added a third run on a ground ball by left fielder Luke Plucheck.
Second baseman Andre Oliver's sacrifice fly to left field gave Prairie View a 1-0 lead in the bottom of the second inning. Catcher Evan Richard reached first on an error then went to third on in inning's second miscue to set up Oliver's RBI fly ball.
Two walks and an error loaded the bases for the Panthers with one out in the bottom of the eighth. Left fielder Darryl Johnson walked to bring in a run and cut the Bearkat margin to 3-2.
Sam Houston got the run back in the top of the ninth as Plumlee led off with a double, was sacrificed to third and scored on a ground ball by Atwood.
With the bases loaded in the bottom of the ninth Burchett got one batter to pop up, struck out the next and got a ground out to end the game.
"I just wanted to throw strikes and get some ground balls," Burchett said. "That strike out for the second out was really a tough at bat. We're just excited to still be playing."
Sam Houston State will play Rice game. The Owls were shut out by Arkansas 1-0 Saturday night.
"Rice and Arkansas are both outstanding teams," Pierce said. "We're just excited to be playing another day. If we can just relax and let the game come to us, we have a chance to compete."
Sam Houston upped its record to 39-21 while Prairie View dropped to 28-25. This was the Bearkats' first NCAA victory at Reckling Park in three regional trips to Houston.