Can enzymes catalyze thermodynamically unfavorable reactions?
Enzymes can catalyze a thermodynamically unfavorable reaction by coupling it with a thermodynamically favorable reaction. Most often, enzymes use ATP hydrolysis reaction (energetically favorable) as a source of energy (in simple terms) to drive the unfavorable reaction forward. One important point to keep in mind here is that enzymes don't drive a reaction equilibrium forward or backward; they just help in achieving the reaction equilibrium faster. Also pay attention that enzyme itself does not change the thermodynamics of a reaction (as you say, ∆G of the concerned reaction remains the same, except at equilibrium), coupling of a favorable reaction with an unfavorable one only helps in making the overall reaction favorable. This phenomenon, of one reaction changing the rate of another reaction, is called induced catalysis and has nothing to do with the enzyme itself.

I will talk about how enzymes do this using examples, while the other answer talks about the semantics. An example which will help you understand this concept is ATP synthase. It is an enzyme found in the inner mitochondrial membrane which allows the movement of H+ from the inter-membrane space towards the mitochondrial matrix, using the energy of movement (since this is favorable and exergonic) to generate ATP. The reaction would look like this:

$$\ce{ADP + P_i + 3 H^+_{IMS} \rightarrow ATP + 3 H^+_{matrix}}$$ (remember that exactly 3 H+ are not required, this is a simplified version)

But, ATP synthase does not drive the reaction forward; it just helps in reaching the following equilibrium faster:
$$\ce{H^+_{IMS} \leftrightharpoons H^+_{matrix}}$$ i.e. keeping the amount of H+ in inter-membrane space and matrix the same. This is why a constant gradient is required to keep them in working condition. For this, the electron transport chain continuously throws H+ into the inter-membrane space to maintain an electrochemical gradient. Along with this, enzymes ATP-ADP translocase and phosphate carrier continuously take out ATP from the matrix and throw in ADP and Pi. However, this lowers the electrochemical gradient (exchanging ADP3- for ATP4- in inter-membrane space, while phosphate carrier catalyzes electroneutral reaction), making the final reaction: $$\ce{ADP + P_i + 4 H^+_{IMS} \leftrightharpoons ATP + 4 H^+_{matrix}}$$

However, since the enzyme just helps in maintaining equilibrium, it can also work backwards i.e. it can also transport H+ against electrochemical gradient (unfavorable), coupling it with ATP hydrolysis (favorable) to reach equilibrium. The enzyme is now called ATPase and the reaction becomes: $$\ce{ATP + 4 H^+_{matrix} \rightarrow ADP + P_i + 4 H^+_{IMS}}$$

There are many more such enzymes which couple ATP hydrolysis (or other favorable reactions) to catalyze reactions which are either unfavorable or very slow (i.e. require a source of activation energy) under normal cellular conditions. Another factor, as the other answer explains, which helps in this is changing the concentration of products. As seen above, ATP-ADP translocase and phosphate carrier continuously decrease the concentration of products (ATP) and increase concentration of reactants (ADP and Pi) so that the overall equilibrium shifts forward and the enzyme keeps on driving the reaction forward (this, again, has nothing to do with with the enzyme itself, it is called Le Chatelier's Principle). Again, the other answer focuses on this issue.

References:


Enzyme - Wikipedia
Cellular Respiration