The result left Igor Cherevchenko's side in third, two points adrift of champions Zenit St Petersburg and CSKA Moscow, who beat Terek Grozny and Anzhi Makhachkala respectively on Saturday to continue their perfect starts.
"It was a very difficult match," said Lokomotiv head coach Cherevchenko.
"We weren't impressive from the start and conceded a silly goal. Luckily, we managed to level after the break."
Left back Grigory Morozov put Dynamo 1-0 up on the stroke of half-time when he headed home Mathieu Valbuena's free-kick from close range.
After the interval, Lokomotiv upped the tempo and were rewarded with the equaliser after 61 minutes.
Alexander Samedov found Croatian international Vedran Corluka in the area and the former Tottenham defender converted unmarked from eight yards.
Dynamo, who were without Russian international forward Alexander Kokorin through injury, have yet to this season but manager Andrei Kobelev remains upbeat.
"It's hard to play without forwards but today we looked the better side," he said.
"We created plenty of chances but scored only once. Unfortunately it wasn't enough to win.
"Of course we would like to win but regardless of the result my team performed well. I think we're moving in the right direction," added Kobelev.
Dynamo, who finished fourth last season, have collected just two points from their opening three league games.
Mordovia Saransk came from behind to secure a point at home to Ural Yekaterinburg thanks to an injury-time equaliser from Yevgeny Lutsenko.
Armenian international forward Edgar Manucharyan's 51st-minute opener had looked like giving the visitors all three points but Lutsenko's late intervention handed Mordovia just their second point of the season.
Meanwhile, Rostov remain unbeaten after a goalless draw at home to FC Krasnodar.