Kallstrom, 31, joined Arsenal on loan on transfer deadline day from Spartak Moscow as cover for the injured Aaron Ramsey and suspended Mathieu Flamini.
However, it has since emerged that the Sweden international is carrying a back injury that is set to keep him sidelined until at least the end of February.
Of the decision to sign the midfielder, Wenger said: “It crossed my mind not to complete the deal and I would not have signed him if we’d had two or three more days to do something else. But it was five o’clock on deadline day, so the decision was: Sign Kallstrom or sign no one.
“He fell on his back two days before he arrived here and the medical showed he had a micro-fracture in his back.
“But many people play with a micro-fracture and don’t even know they have it.
“It is very difficult to identify a player of that stature on a free loan and Spartak have accepted to pay his wages for six weeks.
“It first crossed my mind to sign another midfield player after Aaron Ramsey suffered a setback last Monday. Right across midfield we have had players who have suffered injuries. Mathieu Flamini, Jack Wilshere, Mikel Arteta, Santi Cazorla, Oxlade-Chamberlain — they have all missed part of the season.
“It’s true that we have not sorted that problem for the next four to six weeks.
“But I had to identify a player who has the quality to play for Arsenal and you don’t want to be locked into a deal for three or four years when you keep paying them to do nothing.
“I knew the player well. Kallstrom has 108 caps for Sweden and has played for seven or eight years in France. Yes, it’s possible that he might never play for Arsenal but it’s also possible that he scores a winning goal in a vitally important game.
“We will only know if we made the right decision at the end of the season.”
Wenger’s side returned to the top of the Premier League table on Sunday as they beat Crystal Palace 2-0, and the
French coach has talked up their chances of claiming the title.
He said in The Times: “In the last seven years you have engraved that in your brain and it’s hard to get out now,” Wenger said.
“I can understand that. We haven’t done it [won the title] for a while. We do not play against average teams.
“City have scored over 100 goals, so everyone thinks: ‘Can you beat this team?’ I say yes. It’s all to do. After 24 games we are there.
“It’s now how we finish the season. We gained points from Liverpool and Manchester United this weekend, and we know our results at home will be massively important.”