Using a sample of dilepton top-quark pair ($t\overline{t}$) candidate events, a study is performed of the production of top-quark pairs together with heavy-flavor (HF) quarks, the sum of $t\overline{t}+b+X$ and $t\overline{t}+c+X$, collectively referred to as $t\overline{t}\text{ }+\text{ }\mathrm{HF}$. The data set used corresponds to an integrated luminosity of $4.7\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$ of proton-proton collisions at a center-of-mass energy of 7 TeV recorded by the ATLAS detector at the CERN Large Hadron Collider. The presence of additional HF ($b$ or $c$) quarks in the $t\overline{t}$ sample is inferred by looking for events with at least three $b$-tagged jets, where two are attributed to the $b$ quarks from the $t\overline{t}$ decays and the third to additional HF production. The dominant background to $t\overline{t}\text{ }+\text{ }\mathrm{HF}$ in this sample is $t\overline{t}+\text{jet}$ events in which a light-flavor jet is misidentified as a heavy-flavor jet. To determine the heavy- and light-flavor content of the additional $b$-tagged jets, a fit to the vertex mass distribution of $b$-tagged jets in the sample is performed. The result of the fit shows that $79\text{ }\ifmmode\pm\else\textpm\fi{}\text{ }14\text{ }(\text{stat})\text{ }\ifmmode\pm\else\textpm\fi{}\text{ }22\text{ }(\text{syst})$ of the 105 selected extra $b$-tagged jets originate from HF quarks, 3 standard deviations away from the hypothesis of zero $t\overline{t}\text{ }+\text{ }\mathrm{HF}$ production. The result for extra HF production is quoted as a ratio (${R}_{\mathrm{HF}}$) of the cross section for $t\overline{t}\text{ }+\text{ }\mathrm{HF}$ production to the cross section for $t\overline{t}$ production with at least one additional jet. Both cross sections are measured in a fiducial kinematic region within the ATLAS acceptance. ${R}_{\mathrm{HF}}$ is measured to be $[6.2\ifmmode\pm\else\textpm\fi{}1.1(\text{stat})\ifmmode\pm\else\textpm\fi{}1.8(\text{syst})]%$ for jets with ${p}_{T}>25\text{ }\text{ }\mathrm{GeV}$ and $|\ensuremath{\eta}|<2.5$, in agreement with the expectations from Monte Carlo generators.