St. Louis, four points behind Central Division-leading Detroit, is 0-2-2 over its last four, including a 3-3 tie with Colorado on Friday night.
It was a surprise start for Tkachuk, who missed the previous three games with an infection in his ankle. Weight returned to play after aggravating a groin injury Monday in Detroit and sitting out Tuesday's game in Chicago.
"Of course, it was a lift for us," said Eric Nickulas, who scored a late first-period goal on an assist from Weight. "How could it not be? Tkachuk gutted through with a great performance. No one knew he was going to play."
However, the Blues allowed a late third-period goal to remain winless since a 4-2 victory over San Jose on Dec. 18.
"I thought we were flat and they came at us hard," Blues coach Joel Quenneville said. "They were ready to go."
Columbus has managed just one victory is its last 11 games, and its 25 points are the least in the Western Conference.
The Blue Jackets failed to follow up their first road win of the season on Friday, suffering a 4-3 loss to Dallas the following night.
Rick Nash scored his league-leading 23rd goal and Nikolai Zherdev and Lasse Pirjeta added goals for Columbus, which slipped to 1-9-1 in its last 11.
"We battled back, but I'm getting sick and tired of pulling goaltenders," MacLean said.