United striker Danny Welbeck scored an early opener in the Group B tie, but Taison grabbed a 76th minute equaliser at the Donbass Arena.
Nevertheless, Vidic was pleased with his side's response following a shock home loss to West Brom in the Premier League.
"The most important thing is I saw improvement. I saw solidity, I saw a reaction from the last game," the Serbia international told MUTV.
"We were hard to beat in all positions, not just at the back, but in midfield and wide, and that's the way we have to play. We have to work hard, play as a team and hopefully goals will come."
The 31-year-old continued: "We improved, we didn't let them have many chances to run at our defence. Our team worked really hard but there is still room to improve, and that's a good thing.
"We know we have not played at our best yet, but we still have time to improve and the next game is really important.
"We came to win, we wanted three points but we have to accept as well they were a good team.
"They have quality players and it's hard to play away in the Champions League, but we had some improvement on the last game.
"That's encouraging and we showed character from what happened in the last game, and hopefully we can improve against Sunderland."
The English champions, who are 12th in the domestic standings, next take on bottom-placed Sunderland in a League tie at the Stadium of Light on Saturday.