The Cougars evened their home record to 10-10 with the win. SIUE is 16-26 overall. Quincy fell to 22-25 overall.
"It's very difficult for someone to beat you if you throw strikes, you make all the plays on defense and you get timely hitting," SIUE Head Coach Tony Stoecklin said. "We got all of that tonight."
Stewart was 3 for 4 with the pair of RBIs coming on a single with the bases loaded in the seventh. He also stole two bases and scored a run. Green finished with a single and a double. He walked with the bases load in the seventh and drove in another run with a groundout in the eighth. Green also stole two bases. Alec Saikal and Devin Caldwell each had two hits for SIUE. Caldwell drove in a run with double in the third inning.
"Jacob Stewart swung the bat extremely well tonight," Stoecklin said. "Saikal and Caldwell scalded the ball a couple of times as well. Chase came up big with a couple of RBIs tonight."
Mike Miller (1-3) earned the win tossing a career-high 6 1/3 innings. He allowed a pair of runs in the first inning before settling down. He struck out two.
"After the first inning Mike setted down and gave us 6 1/3 quality innings," Stoecklin said. "Early in games he's a little erratic with his command. Once he works himself through that he settles in and generally throws really well."
Ari Harris was 3 for 4 to lead the Hawks. Jason O'Brien (0-5) started and allowed four runs on eight hits in the loss.
SIUE returns to Ohio Valley Conference play with a weekend series at UT Martin. First pitch Friday night is 6 p.m.