The victory snapped a two-game MLS slide for the Union, who reached the semifinals of the CONCACAF Champions League this month, but were trying to avoid going winless through their initial four league matches for the first time since 2017.
Philadelphia also doubled its goal total from going 0-2-1 over its first three league contests.
Chicago, meanwhile, has dropped three in a row and was shut out for a second consecutive contest. The Fire, sitting on one point, are winless through their first four matches for the first time since 2014.
After a rather pedestrian first half of offensive action from both sides, Philadelphia broke the stalemate during the early stages of the second half. Burke, who initially chested the ball near the Fire net to Jamiro Monteiro, got it back from Monteiro, and deposited his attempt into the top right corner of the net in the 51st minute.
It was nine minutes later when Burke delivered a pin-point cross into the box that Glesnes headed in for the two-goal Philadelphia advantage. The Union have won two of their past three trips to Chicago after losing four straight on the road in the series.
During the first half Saturday, Philadelphia applied most of its pressure over the first 20 minutes. The Fire, meanwhile, picked up its offensive presence in the final portion of the first half. Chicago ended the first half with a 2-1 shots-on-goal advantage. Both teams, which combined for four goals prior to this contest, also had just three corner chances as a collective through the first half.
Philadelphia outshot Chicago 15-9 overall on the match, and 5-3 in terms of chances on target. The Fire's best chance of the match came in the final minutes of extra time when Union keeper Andre Blake stopped a point-blank chance from Nnamdi Chinonso Offor.