Yaks came in many variants - there were Yaks armed with 2 or even 3 20mm in the nose. The aerodynamics of the plane didn't change as the cannon were hidden under the cowling, the weight increase was not that great either, certainly, the overhead was not nearly as great as addition of two underwing cannon pods for 109s.
The design by itself was better than that of 109s in the terms of handling characteristics, visibility from cockpit, speed and climbrate below 18k and potential for modernization. While newer 109 variants only made that aircraft more sluggish and decreased its thrust-to-weight ratio the opposite was observed in the Yak product line. It's not surprising as the Soviet aerospace engineering produced some of the best aircraft designs in preceding 30-s, the first in the world all-metal monoplane fighter I-16, the first in the world truly strategic bomber aviation (TB-3s). An impressive number of world records were set. If it hadn't been for Stalin's paranoia and the purges of 1937, that momentum wouldn't have to be regained at great cost in the early 40-s.
It's a myth that Soviet fighter aviation won because of greater numbers. For example, during air offensive on Moscow, Nazi Germany had numerical superiority and qualitative superiority (VVS had mostly MiG-3s and I-16s at that time), yet that air battle was lost by Luftwaffe.
As a matter of fact, if you read memoirs of the fliers themselves you will see that the opposite was quite often true. 2nd top ranking Allied ace, Alexandr Pokryshkin, mentions many times that he was asking HQ to increase the number of planes dispatched on CAPs from 4 to at least 8 or 12 during Kuban battles of spring of 1943 where his 16th GIAP often had to fight numerically superior flights of 109s.
I have seen a great number of descriptions of documented engagements in which Soviet fighter pilots won while being outnumbered.
Hopefully, more and more of them will be published in English to decrease the level of ignorance some westerners have about that war.