Sullivan added Malkin is expected to miss "a little bit longer" than Bjugstad, but Malkin's injury is not season-threatening.
Bjugstad and Malkin had exited prematurely from the Penguins' 7-2 win, their first victory of the young season, over Columbus on Saturday and did not return.
Malkin was injured in the second period when crashing into the boards with Columbus' Vladislav Gavrikov.
Following the game, Sullivan told reporters the two players were being evaluated but didn't disclose anything more.
Neither player practiced Monday.
Malkin ranked fourth on the Penguins last season with 72 points. The 2011-12 NHL MVP scored 21 goals and provided 51 assists across 68 games. Malkin has been a mainstay in Pittsburgh since the franchise drafted him second overall in 2004.
Bjugstad, meanwhile, arrived in Pittsburgh through a trade from the Florida Panthers in February that also brought the team Jared McCann. Once joining the Penguins, the 27-year-old appeared in 32 games and scored 9 goals while assisting five.
In Bjugstad's absence, Mike DeFabo of the Pittsburgh Post-Gazette pegged McCann as the man to watch. McCann scored two goals and recorded an assist in Saturday's game.
"Nobody can replace [Malkin]," Penguins captain Sidney Crosby told DeFabo after the game. "But when guys go down, other guys have to step up. [McCann] did a good job of that tonight."
McCann appeared in 32 games for Pittsburgh last season, scoring 11 goals and assisting six.
The 1-1 Penguins' first challenge without Bjugstad and Malkin will be at home Tuesday night against the Winnipeg Jets.