Influence on what, exactly? It's hard deny its influence on European civilization, if anything because of thr sheer amount later people intentionally identifiyng with the Greco-Roman tradition. Any influence on a larger scale is terribly hard to quantify or compare.

I don't think it is overstated, the Roman Empire had an enormous impact on so many aspects of the various civilizations that came after it, everything from language, art, law, commerce, governance, engineering, religion, warfare, architecture the list goes on. It would take several books to recount all the ways Rome influenced the world that came after so let me briefly focus on one. The reason Mosques have domes is because in about 126AD the Emperor Hadrian re-built the temple to all the gods and crowned it with the largest unsupported dome ever built, five hundred years before Muhammad was born. The temple Hadrian built is of course the Pantheon which still stands in Rome. The Pantheon was the grandest and most important religious building in Rome after it was built, it's dome was and still is an engineering and architectural marvel. Fast forward to to the year 532, the capital of the Empire has moved from Rome to Constantinople the religion has changed from pagan to Christian and another emperor wants to build (or re-build) the most important religious building of his day. The Emperor was Justinian and the building he erected was the Hagia Sophia. To crown his building he decided to harken back to the pantheon only on a larger more impressive scale. The Haglia Sophia's and its dome went on to influence other churches in the byzantine empire like Kathisma in Bethlehem. starting in the 7th century the Byzantine Empire began it's slow decline, Islamic conquerors began to chip away at Byzantine territory and in the process converted many christian churches to Mosqes, they were particularly impressed with the Byzantine domes and used them as a blue print for their most important Religious buildings like the dome of the Rock on the site of the old temple mount in Jerusalem. When the Byzantine Empire finally fell to the Ottomans in 1453 they converted Hagia Sophia into a mosque and it again influenced the construction of other Mosques and religious building in the Islamic world. Even the Taj Ma Hal in India can trace its architectural lineage back to Hadrian's pantheon. Again this is a very brief explanation and the overall shape and construction of domes changed a lot in that time but it dose not negate the fact that the decision of a Pagan Roman emperor to build his most important religious structure with a dome had a direct influence on the religious buildings of a culture that we might not at face value think have anything at all in common with the likes of Jupiter, Mars, and Venus.