Stephan El Shaarawy and Bryan Cristante put Roma, who lost to AC Milan before the international break, firmly in command by the half-time interval.
But Chievo, beaten 6-1 by Fiorentina last time out and deducted three points by the Italian football federation for false accounting, battled back in the second half.
Valter Birsa pulled a goal back after 52 minutes with Polish striker Mariusz Stepinski sealing a point for the club from Verona seven minutes from time.
"This was a game we had under control, we should have closed it down. This can't happen against Real," Roma coach Eusebio Di Francesco told broadcaster DAZN.
"In games like this you have to score four goals. This didn't happen, and it kept Chievo alive."
The Roma coach added: "We'll be going out to challenge Real Madrid on Wednesday. We mustn't lose our heads, even if these results destabilise us."
Roma are now seventh with five points from four games -- four points behind champions Juventus, who host Sassuolo later Sunday hoping Cristiano Ronaldo can finally break his club scoring duck.
Chievo are propping up the table on -1 points after their deduction during the week.
"We're on the right path," said Chievo coach Lorenzo D'Anna.
"We feared it would be a repeat of the Fiorentina game, but instead we kept our composure.
"We even came close to winning in the end."
Last season's Serie A runners-up Napoli got back to winning ways on Saturday with Lorenzo Insigne scoring the only goal against Fiorentina ahead of next week's Champions League trip to Red Star Belgrade.
Inter Milan's woes were compounded with a shock 1-0 defeat to Parma ahead of their European opener against Tottenham in the San Siro.
In the Stadio Olimpico, Italy winger El Shaarawy got Di Francesco's side off the mark after ten minutes with a volley off Alessandro Florenzi's precise freekick.
On-loan Atalanta midfielder Cristante then turned in his first goal for Roma after half an hour, from distance, after good work from Edin Dzeko.
Stepinski and Birsa both headed over before the break as Chievo dug deep.
Slovenian Birsa found a way through with a curling effort seven minutes after the break, with Stepinski breaking through the tired Roman defence.
Roma goalkeeper Robin Olsen did well to clear a final Emanuele Giaccherini effort to secure a point for the capital side.