In a word, no.

First we need to be clear about the structure of the mujahedin. Most of the mujahedin resistance was Afghan; the foreign fighters, or "Afghan Arabs," were a relatively small and largely inconsequential group within the broader mujahedin resistance. Moreover, the individuals that would come to make up al-Qaeda were a minority within this minority. In other words, we're looking at a really small sub-set of the entire mujahedin. Estimates of the number of Afghan Arabs that arrived in Afghanistan range from 8,000 to 25,000, although the majority of this number actually arrived after the Soviet Union had withdrawn. In the early years of the conflict, the Afghan Arabs tended to fight alongside Afghan resistance groups, and Afghans viewed these foreign fighters with a degree of both respect and trepidation. Afghans were simultaneously impressed by the ideological zeal of the foreign fighters, but also wary of their religious beliefs and ideological justifications for joining the resistance. By 1986, the Afghan Arabs were increasingly closing themselves off from the broader mujahedin resistance movement, creating their own training camps and command structures. In general, these foreign fighters were of so few numbers that they had little impact on the Soviet-Afghan War. (It is also worth pointing out that it's debatable whether the Soviet-Afghan War contributed to the fall of the Soviet Union, or whether the Soviet defeat was caused by an already collapsing political system.

The Afghan Arab resistance to the Soviet Union was insignificant to the Soviet-Afghan War, but it was vitally important to the development of al-Qaeda in the mid-to-late 1990s. It instilled in this small group of foreign fighters a global ideology and a strong sense of camaraderie. This is the historical context into which we should read Zawahiri's statements about how the Afghan Arabs supposedly defeated the Soviet Union. At the time that Zawahiri was writing this stuff about how the Afghan Arabs came together to defeat the USSR, he was explicitly trying to establish a mythological account of al-Qaeda's origins that would both draw new recruits and justify the organization's belief that jihad should go international, mainly by attacking the United States. Before the mid-1990s, the broad jihadi movement was focused on what's called "the near enemy," namely those regimes in the Middle East that the jihadists (justifiably) believed were oppressive and un-Islamic. But al-Qaeda offered a completely different interpretation of jihad, one focused on attacking "the far enemy"--the United States. Most jihadist groups didn't like this turn toward the United States, believing that if the U.S. was attacked, the response would destroy the entire jihadist movement. (This was proven right after 9/11, and explains why so many extremist groups were angry at bin Laden for conducting the attack). This shift to attacking the far enemy needed some justification, and what better way to give it a sense of legitimacy and plausibility than by crafting a narrative that al-Qaeda had already beaten one superpower, so they could easily beat another one as well.

Edit: As was pointed out below, I appear to have misunderstood the question. I'll leave what I wrote and respond here with an answer that more directly ties to the question.

I would be hesitant to say that the Soviet defeat in Afghanistan meaningfully contributed to the fall of the USSR; rather, it was only the latest outcome in a process of Soviet decline that was already underway. The Soviet economy was lagging and the international influence of communism had diminished greatly by the early 1980s. Gorbachev was well aware that Soviet communism as it had been practiced for the last four decades was no longer tenable. In response, he carried out a series of liberal reforms under the programs of glasnost (openness) and perestroika (rebuilding). For Gorbachev, the Afghanistan War threatened his ability to effectively carry out his reforms. Domestically, Soviet citizens believed that their government was wasting resources on unnecessary foreign interventions while they suffered through economic hardships at home. Ironically, Gorbachev's glasnost reforms meant greater freedom to critique Soviet policy, thereby increasing the pressure on Gorbachev to withdraw. Even without the domestic criticism, Gorbachev would have likely still withdrawn as the war exacerbated the problems of an already weak economy hampered by declining prices in key Soviet export commodities. Moreover, most historians agree that, if the Soviet Union had continued the war, it could have withdrawn in a much more favorable position within a couple of years. Gorbachev, however, wanted to get out much quicker in order to ensure that his social and economic reforms could be instituted effectively. In other words, it is probably most accurate to say that the Soviet-Afghan War accelerated, but certainly didn't in any significant way cause the collapse of the Soviet Union.

Sources:

Alex Strick van Linschoten and Felix Kuehn, An Enemy We Created: The Myth of the Taliban-Al Qaeda Merger in Afghanistan.

Fawaz Gerges, The Far Enemy: Why Jihad Went Global

Odd Arne Westad, The Global Cold War