Background/Objectives: The increasing occurrence of extended-spectrum β-lactamase (ESBL)–producing Enterobacterales, most commonly Escherichia coli, has become a serious problem. The aim of this study was to determine the presence of ESBL-producing Gram-negative bacteria in dairy cattle, goat and sheep farms located in southern Türkiye. Methods: Samples (409 quarter milk samples and 110 fresh faecal samples from cattle, 75 bulk tank milk samples and 225 rectal swab samples from goats and sheep) were subjected to selective isolation on MacConkey agar with ceftazidime (2 µg/mL). Isolates were identified by MALDI-ToF MS. The antimicrobial susceptibility profile of the isolates was determined by the broth microdilution method. To obtain a deeper insight into the genetic diversity of isolates substantially contributing to an efficient spread of their ESBL-determinants (23-MO00001: an E. coli from mastitis and 23-MO00002 Citrobacter freundii), the transmission potential and the genetic background of the plasmid carrying the blaCTX-M determinant was studied with whole genome analysis using Illumina sequencing. Results: Of the samples tested, 47 from the bovine faecal samples, 1 from the subclinical mastitis milk sample, 9 from the goat/sheep rectal swab samples and 5 from the goat/sheep bulk tank milk samples had ceftazidime-resistant Gram-negative strains with the ESBL phenotype. Of the 33 ESBL-producing E. coli isolates, 66.6% were resistant to tetracycline, 57.6% to sulfamethoxazole, 48.9% to nalidixic acid, 42.4% to ciprofloxacin and 33.3% to trimethoprim. Pulsed field gel electrophoresis (PFGE) results showed that the majority of E. coli isolates (16/33) and all Enterobacter spp. isolates (n = 5) were not clonally related (80% similarity cut value). The sequenced strains were observed to efficiently transfer their ceftazidime resistance to the recipient strain E. coli J53 at 37 °C (transfer rates: 101–102 transconjugants per donor cell). S1-PFGE showed that the transconjugants J53(p23MO01-T1) and J53(p23MO02-T1) had acquired plasmids of about 82 kb and 55 kb plasmids, respectively. According to WGS results, the E. coli isolate was assigned to ST162, while the C. freundii isolate was assigned to ST95. Conclusions: This study demonstrates that dairy animals are reservoirs of ESBL-producing bacteria.