Burch is suspended for the second leg of the round of 16 series in Guadalajara on Tuesday, and D.C.'s inability - whether playing with 11 or 10 to suppress the Guadalajara attackers along with his absence makes Coach Tom Soehn's selection of players and tactical decisions even more crucial.
Burch's emergence at left back put the final piece in Soehn's switch to a 4-4-2 formation
Both Olsen and Simms scored from distance after Guadalajara had failed to clear balls out of its defensive third, but it the Mexicans should have punished some very shaky D.C. defending much more harshly than they did.
Keeper Troy Perkins saved five shots and watched at least a half-dozen others zoom over the crossbar. Bryan Namoff and Olsen blocked goal-bound shots; any one of them could have been deflected past Perkins but on this night some luck did bounce United's way.
Olsen scored from 25 yards midway through the first half by trapping a poor clearance and teeing off a shot that skipped off the recently sodded RFK infield and bounced past Guadalajara keeper Luis Michel. D.C. had rarely crossed midfield at that point in the match and Burch received his first caution for a reckless foul.
Perkins dropped a cross but Namoff cleared as D.C. invented ways to survive roiling goalmouth scrambles.
Luciano Emilio nearly stole a second United goal five minutes before halftime when he snuck in behind the Guadalajara defense only for his shot to be blocked bravely by Michel.
Just before halftime, Burch arrived a moment late as Omar Esparza hit a cross and appeared to stop as Esparza followed through and their legs clashed. Esparza hit the ground in some pain and Argentine referee Hector Baldaffi brought out the yellow card and then the red.
D.C. had the halftime interval to reorganize but started the second half in complete disarray, with Devon McTavish, Greg Vanney and Namoff ostensibly playing a three-man back line that occasionally Olsen supplemented as right back.
Perkins saved from Alberto Medina as United tried to sort things out and the first time it managed to keep possession in the opponent's defensive third, 10 minutes into the second half, it scored.
Christian Gomez, who labored through a quiet game, bravely battled long enough against three opponents to win a free kick. He crashed his shot against the defensive wall and it rolled into a yawning gap Simms quickly filled to belt a low shot the screened Michel couldn't reach.
Up two goals, still United couldn't sort out its formation, nor hold the lead for longer than six minutes. Josh Gros came on for Jaime Moreno to play right back, but McTavish couldn't fill Burch's spot on the opposite side.
Xavier Ivan Beaz glided down that flank to hit a low cross to the near post for Santana to slam past Perkins in the 61st minute.
After a dozen more minutes of heroic if chaotic defending, Brian Carroll replaced Gomez, who had cramped up twice in the second half. Carroll joined the legion of the desperate: scraping away 50-50 balls, blocking shots and crosses, hoofing high-altitude clearances.
In the final minutes, Perkins slapped away a cross, Olsen repelled a shot from Omar Bravo, and then Gonzalo Pineda wound up his potent left leg to drill a low shot directly at the D.C. keeper. As he gathered it he fell to his knees, as if in gratitude.
United hosts Toronto Saturday in a league match before heading to Mexico.
Soehn can revert to the 3-5-2 favored by his predecessor, Peter Nowak, or insert a left back in place of Burch and hope for the best.
When it played Guadalajara in the CONCACAF Champions Cup' six months ago, it tied, 1-1, at RFK but played better in Estadio Jalisco despite losing, 2-1, and falling 3-2 on aggregate.
This time, D.C. heads to Mexico with a slim one-goal edge, but several defensive and tactical issues to resolve.
Round of 16, first leg
D.C. United 2 Guadalajara 1.
Goals: Olsen 23, Simms 1 55; Santana 1 (Baez ) 61.
D.C. United -- Perkins, Namoff (Boswell, 88), McTavish, Vanney, Burch, Olsen, Gomez (Carroll, 73), Simms, Fred, Emilio, Moreno (Gros 56).
Guadalajara -- Michelle, Magallon, Rodriguez, Esparza (Baez 46), Mejia, Araujo (Padilla, 75), Morales (Arellano 56), Pineda, Bravo, Medina, Santana.
Yellow cards: D.C. United - Burch 23, Burch 44, Carroll 78; Mejia 79.
Red card: D.C. United - Burch 44.
Referee: Hector Baldaffi