Zaccagni broke the deadlock from the spot six minutes after half time after Roma's new signing Dean Huijsen had fouled Taty Castellanos.
Roma came close to securing a late equaliser, but Lazio goalkeeper Christos Mandate made an impressive save from Andrea Belotti's half-volley.
Late in added time, tensions rose as Lazio's Pedro received a second yellow card following an altercation with Leandro Paredes and Roma's Sardar Azmoun was shown a red card for striking Nicolo Rovella.
Roma's Gianluca Mancini was shown a red card for dissent after the final whistle.
