A flow-through biosensor system for the determination of uric acid was developed on the platform of flow-through electrochemical cell manufactured by 3D printing from poly(lactic acid) and equipped with a modified screen-printed graphite electrode (SPE). Uricase was immobilized to the inner surface of a replaceable reactor chamber. Its working volume was reduced to 10 μL against a previously reported similar cell. SPE was modified independently of the enzyme reactor with carbon black, pillar[5]arene, poly(amidoamine) dendrimers based on the p-tert-butylthiacalix[4]arene (PAMAM-calix-dendrimers) platform and electropolymerized 3,7-bis(4-aminophenylamino) phenothiazin-5-ium chloride. Introduction of the PAMAM-calix-dendrimers into the electrode coating led to a fivefold increase in the redox currents of the electroactive polymer. It was found that higher generations of the PAMAM-calix-dendrimers led to a greater increase in the currents measured. Coatings consisted of products of the electropolymerization of the phenothiazine with implemented pillar[5]arene and PAMAM-calix-dendrimers showing high efficiency in the electrochemical reduction of hydrogen peroxide that was formed in the enzymatic oxidation of uric acid. The presence of PAMAM-calix-dendrimer G2 in the coating increased the redox signal related to the uric acid assay by more than 1.5 times. The biosensor system was successfully applied for the enzymatic determination of uric acid in chronoamperometric mode. The following optimal parameters for the chronoamperometric determination of uric acid in flow-through conditions were established: pH 8.0, flow rate 0.2 mL·min−1, 5 U of uricase per reactor. Under these conditions, the biosensor system made it possible to determine from 10 nM to 20 μM of uric acid with the limit of detection (LOD) of 4 nM. Glucose (up to 1 mM), dopamine (up to 0.5 mM), and ascorbic acid (up to 50 μM) did not affect the signal of the biosensor toward uric acid. The biosensor was tested on spiked artificial urine samples, and showed 101% recovery for tenfold diluted samples. The ease of assembly of the flow cell and the low cost of the replacement parts make for a promising future application of the biosensor system in routine clinical analyses.