We report the first experimental demonstration of efficient and high-power operation of a Ti:sapphire laser pumped by a simple, compact, continuous-wave (cw) fiber-laser-based green source. The pump radiation is obtained by direct single-pass second-harmonic-generation (SHG) of a 33-W, cw Yb-fiber laser in 30-mm-long MgO:sPPLT crystal, providing 11 W of single-frequency green power at 532 nm in TEM00 spatial profile with power and frequency stability better than 3.3% and 32 MHz, respectively, over one hour. The Ti:sapphire laser is continuously tunable across 743-970 nm and can deliver an output power up to 2.7 W with a slope efficiency as high as 32.8% under optimum output coupling of 20%. The laser output has a TEM00 spatial profile with M2<1.44 across the tuning range and exhibits a peak-to-peak power fluctuation below 5.1% over 1 hour.