This paper studies the problem of construction of optimal quadrature formulas in the sense of Sard in the $$W_2^{(m,m-1)}(0,1)$$ space. Using the Sobolev’s method we obtain new optimal quadrature formulas of such type for $$N+1\ge m$$ , where $$N+1$$ is the number of the nodes. Moreover, explicit formulas of the optimal coefficients are obtained. We investigate the order of convergence of the optimal formula for $$m=1$$ and prove an asymptotic optimality of such a formula in the Sobolev space $$L_2^{(1)}(0,1)$$ . It turns out that the error of the optimal quadrature formula in $$W_2^{(1,0)}(0,1)$$ is less than the error of the optimal quadrature formula given in the $$L_2^{(1)}(0,1)$$ space. The obtained optimal quadrature formula in the $$W_2^{(m,m-1)}(0,1)$$ space is exact for $$\exp (-x)$$ and $$P_{m-2}(x)$$ , where $$P_{m-2}(x)$$ is a polynomial of degree $$m-2$$ . Furthermore, some numerical results, which confirm the obtained theoretical results of this work, are given.