Aims: The neutron star low-mass X-ray binary GRS 1741.9-2853 is a known type-I burster at the Galactic center. It is transient, faint, and located in a very crowded region, only 10' from the supermassive black hole Sgr A^star. Therefore, its bursting behavior has been poorly studied so far. In particular, its persistent emission has rarely been detected between consecutive bursts, because of lack of sensitivity or confusion. This caused GRS 1741.9-2853 to be one of the nine "burst-only sources" identified by BeppoSAX a few years ago. The physical properties of GRS 1741.9-2853 bursts are of significant interest beacause we know little about the nuclear regimes of low accretion rate bursters. For the first time we analyze several bursts along with the persistent emission of the source, using INTEGRAL, XMM-Newton, and Swift observations. Methods: We investigate the source flux variability and bursting behavior during its 2005 and 2007 long outbursts. These events were monitored almost entirely by INTEGRAL, both in the soft (3-20 keV) and hard (20-100 keV) X-ray bands, and also partly by XMM-Newton and Swift (2-10 keV) in 2007. Results: In its last activity periods, 2005 and 2007, the persistent luminosity of GRS 1741.9-2853 varied between 1.7 and 10.5×1036 erg s-1, i.e., 0.9-5.3% of the Eddington luminosity. The shape of the spectrum as described by an absorbed power-law remained with a photon index Γ ≈ 2 and a column density N_H≈ 12×1022 cm-2 throughout the outbursts. We discovered 11 type-I bursts with INTEGRAL, and inspected four additional bursts: two recorded by XMM-Newton and two by Swift. From the brightest burst, we derive an upper limit to the source distance of 7 kpc. The observed bursts characteristics and source accretion rate infer pure helium explosions igniting at column depths y_ign≈ 0.8-4.8 × 108 g cm-2, for typical energy releases of 1.2-7.4×1039 erg.