The Belgian, who scored 23 goals last season, has not found the net for three months and was left out of the squad altogether by manager Paul Lambert for Saturday's 2-1 defeat by Stoke.
Benteke has struggled with various niggles this season and Lambert feels he needs time to fully get over a knee problem suffered on international duty.
Villa have managed only 17 goals in 17 Barclays Premier League games this season and have failed to score in three of their last five matches.
Weimann and Libor Kozak bore the goalscoring responsibility at the Britannia Stadium with Benteke absent and Gabriel Agbonlahor suspended, and it was Kozak who seized on a misjudged back header from Erik Pieters to score.
That cancelled out Charlie Adam's 51st-minute opener but parity lasted just four minutes before Peter Crouch knocked in the winner in the 70th minute.
Weimann said of Villa's goal record: "I wouldn't say it's a problem, we're still scoring goals, but obviously Christian scored loads of goals for us last year. This year other people have got to come in and score goals.
"Everyone talks about the fact Christian hasn't got a goal for a bit but once he scores he'll be all right. We're still scoring from other areas, not just Christian.
"He's a big player for us so we obviously miss him but Libor came in and played really well. He got another goal and he looked really good. He held the ball up really well and you can't really ask any more of him."
Weimann has also found goals hard to come by this season, managing only two and none since September.
"Obviously I'd like to score a few more goals but I'm just trying to do my best and hopefully I can get another goal soon," said the Austrian.
"We're still playing attacking football and it's just not come off in the last few games but I'm sure we'll get a few more goals."
For Stoke, the result gave them a second home league win in succession following the shock victory over Chelsea two weeks ago.
They have lost only once in the league in two months and moved into the top half of the table.
Full-back Geoff Cameron said: "It's huge, it's a confidence builder. It was a big game and a big three points. We're trying to create a gap behind us and trying to close the gap in front of us.
"Besides the Capital One Cup, we're one loss out of eight games. We're trying to find some form and, even though it wasn't pretty, we got the job done, which is a good sign.
"You want to take advantage of the home games and get as many points as you can."
Crouch was named man of the match but Cameron, who has impressed since joining the club from Houston Dynamo in the summer, played a big part in both goals.
His deep free-kick allowed Crouch to flick on for Adam, while it was Cameron's cross that caught out Brad Guzan and Chris Herd to allow Crouch a tap-in.
Cameron almost opened the scoring himself in the first half with a cross that would have dipped under the bar had his USA international team-mate Guzan not got a hand to it.
"It was a decent cross," Cameron said. "I would have loved that to go in on Brad, he would have never heard the end of it.
"I had a few good crosses. You just want to put the ball in the box and create opportunities for one of the guys to run onto.
"(For Crouch's goal) I looked up and there was some space between the goalkeeper and the back four so I just put it behind them. Jon (Walters) made a great run, he just missed it, then Crouchy comes in and puts it in the back of the net. It was huge for us.
"Their goal was unlucky for Erik, it would have been a smart play defensively but he didn't see the guy. It was unfortunate but we fought back. I have to pay tribute to the guys, they left it all on the field."