Coach Eric Shore's Tigers, who are ranked 17th nationally, finished their season with an 19-9 record. Illinois, which is ranked 16th, improved its record to 19-7.
Tied at 3-3, the match was decided at No. 3 singles with Illinois junior Bruno Abdelnour taking a 7-6, 2-6, 6-3 win over sophomore Daniel Cochrane.
"There were a couple of things in that match with Dan that you can't control," Shore said. "It's tough on the road when it comes down to crunch time, and things don't really go in your favor. But that's the way it is.
"The NCAA committee felt like we shouldn't have hosted, and that's a hard one to swallow right now. I don't think we lose this match at Auburn. Illinois fed off their crowd a little bit at the end, and we would have fed off our crowd at home. But we put ourselves in a position to win, and you can't ask any more than that. I'm proud of them."
The Tigers took victories with Alex Stamchev winning at No. 2 singles, Lucas Lopasso winning at No. 4 singles and Lukas Ollert winning at No. 6 singles.
Stamchev, the winningest player in Auburn history, won 7-6, 6-3 over Roy Kalmanovich to tie the match at 2-2 after the Tigers trailed 2-0. It was the senior's 105th career singles victory and 212th victory counting doubles.
Lopasso took a 6-4, 7-6 victory over Tim Kopinski that put the Tigers ahead 3-2.
Dennis Nevolo, who is ranked No. 7 nationally, pulled Illinois back to even at 3-3 with a 7-6, 6-4 victory over No. 18 ranked Andreas Mies at No. 1 singles.
Ollert was the first Auburn player to win on Saturday as he took a straight sets victory by scores of 6-3, 6-2 over Ross Guignon.
Illinois won the doubles point to start the match and took a 2-0 lead when Stephen Hoh defeated Dennis Lengsfeld 6-2, 6-3 at five singles.
"What can you say about a match like that?" Auburn head coach Eric Shore said. "The guys left it all out there. We got off to a slow start in doubles and really, really battled back. We put ourselves in a position to win. Even though we lost the doubles point, I didn't feel bad. I thought we were carrying some momentum into singles, and we did. Ultimately, we came up a little bit short. I'm proud of these guys and everything they did.
"We've been in tight matches like that over the last few years, but not in that situation, for a Sweet 16 berth," Shore said. "But would you rather lose 4-0 and be done in three hours, or would you rather lose the last match of the day after a five-hour battle? I'll take the five-hour match, because you know you're in the battle. And I thought we proved that today."