Yes, though the improvement is not direct, not large, and not just from classical music.
Listening to any emotionally positive music has been shown to improve a subject's mood, which in turn has been shown to improve cognitive abilities. Furthermore, although positive effects on certain tasks have been observed, the effects appear to be quite small overall.
In short, it's your mood and not the music. If listening to classical music improves your mood, then yes, classical music will have a small positive effect on your performance at various cognitive tasks. If listening to heavy metal improves your mood, then heavy metal will provide the same small performance improvement.
Source of Belief: The idea that classical music leads to a noticeable improvement in cognition can be traced back to a study in 1993, as well as a second study in 1995, whose impressive results helped popularize the idea the classical music improves IQ to findings that listening to a certain Mozart piece temporarily improves spatial reasoning ability. "Mozart temporarily improves spatial reasoning ability" eventually transformed into "classical music improves IQ", and now it's a fairly common belief.
However, as can be see in this paper which recreated the original Mozart experiment, the impressive results could not be replicated:
The effect size for the contrast of Mozart versus silence in Rauscher et
al. (1995) was substantial (d = 0.72), whereas the effect size for the
same contrast in this study was d = 0.06. One would have expected
that fidelity to the procedure of Rauscher et al. would produce a similar
effect size. Chabris (1998) calculated an average effect size of d =
0.16 for all 15 Mozart-versus-silence comparisons published, or submitted,
Positive Music -> Positive Mood -> Improved Performance: Although the impressive results of the original Mozart experiment cannot be reliably replicated, there still seems to be some positive effects of listening to classical music. This paper sought an explanation for the apparent effect, and concluded that cognitive improvements are related to improved mood. They compared the previously mentioned Mozart piece, a 'sadder' piece (Albinoni’s Adagio in G Minor), and no music. The paper had the following results:
Participants performed better on a test of spatial abilities after listening
to a Mozart sonata than after sitting in silence. When a slow,
“sad” musical excerpt by Albinoni was presented instead of a Mozart
sonata, there was no effect of exposure to music... Participants
who listened to Mozart scored significantly higher on positive
mood and arousal (enjoyment rating, mood rating, POMS arousal
score) and significantly lower on negative mood (POMS mood score)
compared with their counterparts who listened to Albinoni. In short,
our findings provide compelling evidence that the “mysterious”
Mozart effect (Steele, Bass, & Crook, 1999) can be explained by participants’
mood and arousal level... As we noted earlier, a large body of scientific evidence confirms that arousal and mood influence performance on a variety of cognitive tasks.
This paper, studying Canadian college students and Japanese 5 year-olds, led to similar conclusions:
Canadian undergraduates performed better on an IQ subtest (Symbol Search) after listening to an up-tempo piece of music composed by Mozart in comparison to a slow piece by Albinoni. The effect was evident, however, only when the two pieces also induced reliable differences in arousal and mood. In Experiment 2, Japanese 5-year-olds drew for longer periods of time after singing or hearing familiar children’s songs than after hearing Mozart or Albinoni...
The results for the 5-year-olds provided additional evidence for the 'better mood->better performance' hypothesis, as listening to songs they enjoyed resulted in longer concentration on a task than either type of classical music.
Size of the Effect: Regardless of the reason behind the effect(direct from music or indirect from mood), the results are quite small. This (unfortunately paywalled and terribly named) meta-analysis examines nearly 40 studies related to the original 1993 Mozart experiment. Their analysis concluded that the effect was both minor and resulted from multiple types of music:
Sixteen years after initial publication we conduct the so far largest, most comprehensive, and up-to-date meta-analysis (nearly 40 studies, over 3000 subjects)... We could show that the overall estimated effect is small in size (d = 0.37, 95% CI [0.23, 0.52]) for samples exposed to the Mozart sonata KV 448... Additionally, calculation of effect sizes for samples exposed to any other musical stimulus and samples exposed to a non-musical stimulus or no stimulus at all yielded effects similar in strength (d = 0.38, 95% CI [0.13, 0.63])...