The title says it all-- Despite typical rumors, bad memories and misconceptions on the theory, Dexter's Laboratory has never crossed over in any actual episode of The Powerpuff Girls, other than in the IDW comic strips.
In January 28, 2015, IDW Publishing published a crossover comic miniseries called "Powerpuff Girls: Super Smash-Up!" As the title suggests, the comic focuses on The Powerpuff Girls's regular "team-ups" with various heroes similar to the most well-known version of the original Cartoon Network series. One pair of non-Powerpuff Girls characters The Powerpuff Girls teamed up with from another show were Dexter and his sister DeeDee of the show "Dexter's Laboratory." This comic strip is available for reading online.
Although a comic strip for The Powerpuff Girls and Dexter's Laboratory's crossover exists under Powerpuff Girls: Super Smash-Up!, Cartoon Network unfortunately has not confirmed an actual episode pairing the two shows together. The comic strip was released a year before The Powerpuff Girls rebooted in 2016 for a new series that is directed by Spongebob Squarepants animation/art director Nick Jennings. So far, the reboot ran for two seasons, but it is unknown if it will continue and leave a spot for a Powerpuff Girls/Dexter's Lab crossover episode to be legally confirmed. However, there are reasons for the two universes of The Powerpuff Girls and Dexter's Lab to meet because they were almost the same creators and producers (Craig McCracken and Genndy Tartakovsky) involved in both of the cartoons. But for some reason, Cartoon Network chose not to confirm the crossover, perhaps due to their policy to not accept unsolicited ideas, and the crossover went unproduced.
A fan by the alias Bigfatloser86 (known as BigRedMonster86 on Wikia) has made fanart of and posted various (failed) crossover ideas of "The Powerpuff Girls" and "Dexter's Laboratory" on deviantART, such as "Go, DD! GO!" and "Diva Dynamite and the Superpuff Girls." He has been in touch with a former animator who worked on The Powerpuff Girls and Dexter's Lab; He asked about producing the crossover idea as a miniseries (specifically created for and during the third season of the 2016 series of The Powerpuff Girls) for Cartoon Network, and the animator said that he believes Cartoon Network's policy won't allow any Powerpuff Girls crossovers, but so far, no physical evidence for that being the reason has been put forward.
So far, the only two legally confirmed crossovers, fully pairing both universes together in the plot, were "Powerpuff Girls: Super Smash-Up!" and "Cartoon Network: Super Secret Crisis War!," both which were only comic strips published by IDW publishing and never made into animated specials. Characters from The Powerpuff Girls and Dexter's Laboratory had cameo appearances in some Cartoon Network episodes or crossover specials such as "The Grim Adventures of KND." Some Dexter's Lab characters (including Dexter) also had cameo appearances in a few original episodes of The Powerpuff Girls during it first four seasons, and one character, Major Glory, fully crossed over in a season 4 episode "Members Only." The remaining Dexter's Lab characters (including DeeDee) have never appeared commercially in any actual episode of The Powerpuff Girls and still remain self-contained, and only time will tell when Cartoon Network decides to confirm a crossover idea fully pairing all characters of the two universes. However, as of now, there's a very small chance Dexter's Lab will ever cross-over with The Powerpuff Girls in any actual episode of the 2016 series, since many Cartoon Network shows are usually like that.
Not too much else is known about the actual/official crossover episode of "The Powerpuff Girls" and "Dexter's Laboratory," including why it was scrapped and if it was finished or not. However, it is more than likely that Cartoon Network had banned DeeDee from crossing over in another show, reportedly due to a number of negative debates about how ugly and/or annoying DeeDee was, which could be why she was the only Dexter's Lab character to be left off The Powerpuff Girls from cameo appearances and/or full crossover appearances. The actual reason DeeDee doesn't crossover with The Powerpuff Girls (in the actual show) is because many older Cartoon Network characters are usually like that, despite The Powerpuff Girls being the most popular Cartoon Network series while Dexter's Lab actually wasn't all that popular. On the other hand, this is exactly why Cartoon Network permanently cancelled Whatever Happened to... Robot Jones? (and also removed it from further reruns in the U.S. after late 2008) because they wouldn't let Greg Miller (the creator of Robot Jones) do whatever he wanted to do, just like how Pixar and Disney were like back in the day.
On deviantART, there is still an ongoing fan production effort for the crossover going on, and it is unknown if the fan crossover will be completed. Only time will tell when Cartoon Network will ever feel like planning an official crossover episode pairing The Powerpuff Girls with DeeDee (or the rest of the Dexter's Lab cast, regularly) if that's possible.