He said a partnership with Putin, "based on the fact that we have long worked together and trust each other" can give the country "rather interesting results".
Signalling that there would be little change in foreign policy, he said his government "should pursue independent foreign policies, the ones we had in the past eight years".
On Sunday, Putin congratulated his protege.
"Our candidate, Dmitry Anatolyevich Medvedev, has taken a firm lead," Putin said, wishing him success as he appeared alongside Medvedev on an open-air concert stage in Red Square, outside the Kremlin.
Al Jazeera's Jonah Hull, reporting from Moscow, said the result was not a surprise and was never really in any doubt.
The poll, he said, was just a formality - a Soviet style vote with really no choice at all.
Even before the election was under way, critics had denounced it as little more than a stage show.
As voting progressed, the independent Russian election monitoring group said it had received a steady stream of complaints.