Ljajic arrived from Fiorentina as a replacement for Tottenham-bound Erik Lamela this week and announced himself with a sensational strike after Miralem Pjanic had followed up an own goal from Fabrizio Cacciatore.
Verona, whose season began with a shock home win over AC Milan, were second-best throughout and had goalkeeper Rafael to thank for keeping the scoreline respectable.
It took Roma three minutes to threaten as Alessandro Florenzi beat the offside trap to shoot wide of Rafael's goal.
Ivorian forward Gervinho, making his full debut, then forced a low save from the Verona keeper and the siege continued as he punched a Pjanic rocket out for a corner.
Verona finally bared their teeth as Luca Toni tried to pull off a backheeled finish from Emil Hallfredsson's pass.
On the half-hour mark, Pjanic seized on a clever backheel from Francesco Totti to pick out the onrushing Florenzi, who missed the mark by mere inches and Kevin Strootman's subsequent snapshot pinged off a stranded Rafael.
Rudi Garcia's men picked up where they had left off after half-time, Totti's ball to Florenzi only staying out of the net thanks to a deflection off Vangelis Moras.
The opener finally arrived in the 56th minute and, although the ball was decisively clipped beyond Rafael by the covering Cacciatore, Maicon celebrated wildly on the Stadio Olimpico's running track.
Totti had picked out the Brazilian full-back's burst up the right wing and the former Manchester City man gleefully smashed the ball off Cacciatore's leg and out of Rafael's reach.
Verona were cowed and Rafael was duly beaten again inside three minutes.
This time the ball's flight was uninterrupted as Pjanic steered a superb lob over the Gialloblu keeper from the edge of the D.
Lost in the festivities had been Ljajic's introduction in place of Florenzi but the Serbian certainly made his presence felt with the third goal of the night in the 66th minute.
It was a third goal from distance, too, as Ljajic loped forward to switch feet and hammer the ball off the inside of a post.
The damage had already been done but Hallfredsson gave the travelling fans something to get excited about when he clattered the crossbar 10 minutes from time.