From onscreen material, there truly isn't anything that actually says or indicates that the Defiant's weapons are more powerful than a Galaxy-class. It's been colloquially referred to as a heavily armed warship and one of the most powerful, but those terms can also be applied to the Galaxy-class or any big starship, really.
The Defiant manages to destroy a Jem'Hadar attack ship in "The Search" with its pulse phasers in a few shots. In comparison, the Odyssey doesn't manage to destroy one with its own phaser fire in "The Jem'Hadar" (although according to the script, a "Jem'Hadar ship is crippled" after exchanging fire with the runabouts Mekong and Orinoco and presumably the Odyssey earlier on). So in terms of phasers, the Defiant had more powerful ones (at the time). In terms of torpedoes, the Defiant had more advanced ones (at the time). Now, a Galaxy-class starship had torpedo tubes that could fire up to at least 5 simultaneously, and they could be used to rapid-fire bursts of torpedoes (like in "The Survivor" or "The Cost of Living"), so the Odyssey might still hold the firepower advantage... but torpedoes probably wouldn't have been effective against the small and nimble Jem'Hadar attack ships.
Now, the two episodes in question occur in late 2370 and early 2371. As aforementioned, the fleet-wide upgrades in mid-2372 seem to narrow (or maybe even eliminate?) whatever firepower advantage the Defiant had in early 2371.
I'd submit that it could beat an Excelsior-class ship in a head-on engagement, though, but I do think that's more of a result of being able to outmaneuver it and deliver more direct hits (its biggest advantage against any larger vessel, IMO).
Leyton expected the upgraded Lakota to defeat the Defiant in battle... if it wasn't for those meddling kids and their fancy ablative armor...

But yeah, maneuverability seems to be a major factor alluded to in "Heart of Glory" and then finally demonstrated in full in DS9. Heck, the Valiant might have survived after damaging the Dominion battleship (as strong as *three* Galaxy-class starships?!) had it just run away (or at least kept zigzagging) when the crew realized that their trick couldn't finish the job.