Martial won and converted a late spot-kick to extend United's lead to 4-0 in the closing stages of the Group H clash at Old Trafford on Wednesday night.
Rashford had initially placed the ball down, only to see his team-mate step up and beat Peter Gulacsi from the spot, but the United forward would not be denied the match ball.
Martial ultimately set Rashford up for his third goal, scored in the second minute of added-on time, as the 22-year-old completed his first career hat-trick.
Solskjaer was delighted with Rashford's performance but had no problem with Martial taking the penalty, as that had always been the plan.
"Before the game I said Antho' was going to take that penalty and that's it," Solskjaer explained. "If you look at the starting squad it was either Antho' or Paul [Pogba] to take it and Paul was taken off."
Rashford said he had spoken to Martial "a little bit" before the kick but was happy to let his team-mate step up. "I feel confident with Anthony taking them as well," he told BT Sport. "So no problem."
Bruno Fernandes, United's regular penalty taker, was also on the pitch when Leipzig's Marcel Sabitzer fouled Martial in the 87th minute.