Citadin Eder had the game's first chance after just two minutes, but he failed to keep his shot on target.
Mattia Destro squandered Roma's first chance before Francesco Totti's cross-shot almost went in past Sergio Romero between the Samp posts, only to hit the crossbar.
Romero denied Erik Lamela's drive with a good save and deflected Alessandro Florenzi's volley wide.
Roma finally breached the Sampdoria citadel in the 34th minute when Destro's shot bounced off Romero and Florenzi's muffled pass fell to Totti, who slotted home from just nine yards.
Samp went down to 10 men a minute after the half-time interval, as Enzo Maresca was dismissed for a second bookable offence.
However, that did not deter the visitors from Genoa from equalising on 62 minutes when Maarten Stekelenburg's failure to clear away a Gaetano Berardi cross saw Gianni Munari tap in from just six yards.
Both Roma and Sampdoria went close to adding to the scoreline in the final stages, but their failure to do so meant that the contest at the Stadio Olimpico ended in a 1-1 draw.