I have been scouring the internet for several hours and cannot find any mention of any specific agency or division of a security agency within the USSR that was specifically meant to guard the Soviet leaders. But I can't imagine that there wasn't one.
The security of Soviet political leaders was the principal responsibility of the KGB (MGB and MVD before 1954). Of course the KGB traced its roots to 1917 as the Cheka was created in response to an assassination attempt on Lenin's life. The Cheka, and later the KGB, featured a ''Sword and Shield'' as their logo to symbolize their role in the Communist system. The KGB was both the shield of the Party and Revolution, shielding it against the counterrevolutionary enemies, and its sword to strike at the class enemy.
In particular, the Protective Service of the KGB (Sluzhba Okhrany), also known as the 9th Department, was responsible for the protection of Party and State leadership figures after 1945. Until 1959, there had also been a separate KGB Department, the 10th, which was a sort of Praetorian Guard - an armed cadre of troops stationed within the Kremlin Walls to keep out armed intruders (including the military if necessary) and to protect the people inside the Kremlin and government buildings. When in 1954 for instance a group of plotters led by Nikita Khrushchev wanted to oust Lavrentiy Beria - the chief of the MVD and ascendant successor to Stalin - they had to figure out how to smuggle weapons into the Kremlin past these guards and arrest - or rather kidnap - Beria without these guards, who were answerable to Beria, intervening to protect him. The plot eventually required the aid of Marshall Zhukov, who not even these Kremlin Guards would dare to search for concealed weapons, and calling away the Kremlin Guard Commander at the latest moment to put him at physical distance from the affair.
The Kremlin Guard was after 1959 placed directly under the KGB's Protective Service, which also featured other offices concerned with personal security (bodyguards), a counterintelligence unit, an automotive unit, and sections looking after the dacha's of prominent figures, and even a branch was dedicated entirely to security at Crimea, where Soviet leaders often came for vacationing or receiving foreign leaders. The ''Spetskukhnya'' of the KGB Protective Service was the KGB's own special cooking club, responsible for the food and drinks served to Soviet leaders and ensuring they wouldn't be poisoned.
The protection of important buildings was after 1969 shared with the new 15th Directorate, which also took responsibility for the protection of important military facilities and vital infrastructure.
On top of all this, the KGB and MVD (Internal Affairs) had some heavily armed paramilitary forces at their disposal in and around Moscow in the event of serious disturbances such as a military coup, mass protests, or terrorist attacks. These included the MVD's Dzerzhinsky Division, but also Alpha Group, created by KGB Order No. 0089 in 1974. Alpha Group specialized in counter-terrorist operations and its creation should be seen in the context of the 1972 Munich Olympics massacre and the emergence of specialized counter-terrorism forces in many countries.
So in short, yes, the Soviet Union had its own equivalent to the US Secret Service, but it was included within the KGB and some of the more paramilitary aspects of protective security were handled by specialized forces borrowed from other institutions such as the MVD which was mostly responsible for regular policing and law enforcement. The KGB was chiefly responsible for the protection of the Communist Party, and the Soviet leadership in particular. To an American, I would perhaps describe the KGB as the equivalent of merging the FBI, Secret Service, CIA, NSA, US Coast Guard, US Customs and probably some agencies I forget in one super agency. Hence, at its peak, the KGB employed nearly 500,000 people. Those were paid, official, employees - not counting recruited agents and informers.