Absolutely. The Revolution and Civil War devastated the old elites and it's safe to say that they comprised a substantial percentage of the 500,000-3,000,000 Russians who emigrated during this period. Those that remained were classified as lishentsy (former people), disenfranchised and subject to persecution and discrimination by the new regime. Yet this category - which included members of the nobility, Tsarist police, White Guardists, priests, businessmen, etc - peaked at some four million souls in the mid-1920s. There were still plenty of traces of the ancien regime left in Soviet Russia.
And while many of the lishentsy lived clandestine existences (with former ladies-in-waiting hiding in theatre choirs), those who could took the most obvious escape route: service with the Soviet state as a 'specialist'.
Members of the old elite were often well educated, skilled, experienced and with a history of service to the state. The Bolsheviks, struggling to retain state power, lacked almost all of these attributes. Intellectuals and workers, they were were entirely new to the business of government, with little grasp of administration. Trotsky, newly Commissar for Foreign Affairs and with responsibility for ending the war, later confessed that his initial plans for the ministry amounted to "issuing a few revolutionary proclamations to the people of the world and then shut up shop."
To actually run their new government the Bolsheviks needed the assistance of Tsarist officials. Once it became clear that Soviet power was here to stay, that was broadly forthcoming. Despite an influx of workers into the state apparatus, and the flight of many former officials abroad or to the White camps, there was significant continuity in personnel between the Tsarist and Soviet states. These 'bourgeois specialists' were able carve out areas of influence within the state and industry.
Let's illustrate with a stat attack: a mid-1918 census of the state bureaucracy established that only 13% of officials were party members or sympathisers, showing the clear limits to Soviet influence in the state apparatus. In 1920 specialists comprised over 75% of VSNKh (Supreme Council on the National Economy) employees. Indeed, across the government as a whole and despite strenuous efforts to promote workers into positions of authority, as late as 1933 only 30% of leading Soviet officials were former workers. Now, not all of the remaining 70% would have been pre-revolution officials but certainly they would have comprised the majority.
Of course this continuity created tensions of its own for a revolutionary party. The Bolsheviks vacillated between condemning specialists in favour of worker promotion/management and, equally ideologically, protecting them as an integral part of the emerging industrial economy. This was a debate that wasn't really settled until the arrival of a new generation of Soviet-trained 'red specialists' in the late 1930s. Until there there would be semi-regular bouts of 'specialist-baiting' and show trials, indicating the deep unpopularity of maintaining former Tsarist officials in positions of authority in the new Soviet world.
The specialists themselves needed also to justify their collaboration, if only to themselves. Members of a privileged class before the Revolution, few were automatically sympathetic to Bolshevik Marxism. Many held their noses and worked further the aims of a hostile regime out of a tradition of serving the nation. Such 'National Bolsheviks' accepted the loss of the Tsar but were often enthused by Bolshevik plans for modernising the country. Russia would be made great again, to borrow the modern lingo, but under a red banner. For example, I talk a bit here about how the grand technical projects of the 1920s appealed to the Russian scientific establishment.
This nationalism held as much appeal for the military, spurred by the threat of territorial dismemberment from abroad. General Parsky, a Red Army commander, admitted that:
I am far from this socialism that your Bolsheviks preach. But I am ready to work honourably not only with them, but with anyone, even the Devil and his disciples, if only to save Russia from German slavery.
The new Red Army relied heavily on such officers from mid-1918. While never an easy relationship, as the commissar system hints at, almost 50,000 former Imperial officers served with the Reds during the Civil War and they came to enjoy a near-monopoly on senior military commands. This represents roughly 30% of the pre-revolution officer corps.
Not all soldiers served willingly, and plenty surely did so just for a salary, but it was possible for Tsarist officials to accommodate themselves in the new Soviet regime. This applied to most fields. The Bolsheviks, with some reluctance and mutual suspicion, relied on the remnants of the old government to build their new state. Tsarist officers, diplomats, economists, scientists, etc were all prominent in 1920s Soviet society. For these officials, becoming a 'specialist' provided a job, some shelter from persecution and the chance to continue their careers serving Russia. As the figures above suggest, the numbers involved were not insubstantial.
Reading
I've drawn from some relatively varied sources here. The key works are probably David Rendle's Defenders of the Motherland and Shelia Fitzpatrick's Education and Social Mobility in the Soviet Union. Some specific stats and quotes came from RW Davies' Crisis and Progress in the Soviet Economy and Evan Mawdsley's Russian Civil War.