Trailing 42-34 with 44 seconds to play and fourth down at the Illini 3-yard line Saluki quarterback Kory Faulkner threw too high for leaping wideout Adam Fuehne.
Illinois was up 42-24 with as little as 12 minutes left.
The Illini offense was mostly Scheelhaase. He was 28 for 36 with one interception, and his second-quarter touchdown passes to Jon Davis and Josh Ferguson built a 25-7 halftime lead.
Ryan Lankford caught six passes for 115 yards and Ferguson had four for another 103 yards on a hot, muggy afternoon.
The temperature hit 95 and the heat index 108. Big Ten officials said one linesman, Bob Davis, was hospitalized for heat exhaustion. His condition wasn't immediately available but a conference spokesman said Davis was alert and speaking with doctors just after the game.
Beyond Scheelhaase and the Illinois passing attack, there had to be cause for concern among the Illini.
The Salukis, a six-win FCS team in 2012, struck easily against a young Illinois secondary. Faulkner was 25 for 40 for 312 yards and three touchdowns, and was never sacked or even heavily pressured.
And the Illini offensive line had a rough afternoon. Illinois had just 49 rushing yards on 30 carries against a small Southern Illinois defense, and the Salukis had five sacks.
After falling behind 7-3 and sending a wave of anxiety through the crowd of 42,000, Scheelhaase' first touchdown pass put the Illini up to stay. He tossed a short pass to Ferguson, who weaved past a Saluki defender to the right sideline for a 53-yard score and a 10-7 lead with 9:31 left in the first half.
That opened a nine-minute span that was all Illinois.
The Salukis barely had time to get their offense back on the field before Illini tackle Tim Kynard hammered running back Mika'il McCall as he came through the line, jarring the ball loose for Illinois' Houston Bates to dive on.
With 9:02 left in the half, Donovonn Young plunged into the end zone from two yards out for a 17-7 lead.
Then, after Southern Illinois elected to punt rather than go for it on a fourth-and-1 at the Illinois 43, the Illini too the ball at their own six with 1:32 left in the half and drove deep into SIU territory. From the Saluki 11, Scheelhaase hit Davis on the right sideline. Davis stumbled to the end zone, diving to push the ball in, for a 25-7 lead.
Illinois built that lead to 39-17 with just over five minutes to play in the third quarter on a 100-yard kickoff return by V'Angelo Bentley. He tied a school record for longest kickoff return and seemed to deliver a blow that might finish the Salukis.
But just over two minutes later Faulkner hit wide receiver John Lantz for a 35-yard touchdown with to pull to within 39-24. The teams the traded field goals to make the score 42-27.
Faulkner brought the Salukis to within eight at 42-34, this time hitting tight end Mycole Pruitt from a yard out.
Pruitt finished with five catches for 83 yards and the touchdown. Lantz caught four balls for 99 yards and one touchdown.
Illinois hosts Cincinnati on Saturday while the Salukis will face Eastern Illinois at home.