We investigate the fraction of close pairs and morphologically identified mergers on and above the star-forming main sequence (MS) at 0.2 ≤ z ≤2.0. The novelty of our work lies in the use of a non-parametric morphological classification performed on resolved stellar mass maps, reducing the contamination by non-interacting, high-redshift clumpy galaxies. We find that the merger fraction rapidly rises to ≥70 per cent above the MS, implying that - already at z ≳ 1 - starburst (SB) events (∆MS ≥ 0.6) are almost always associated with a major merger (1:1 to 1:6 mass ratio). The majority of interacting galaxies in the SB region are morphologically disturbed, late-stage mergers. Pair fractions show little dependence on MS offset and pairs are more prevalent than late-stage mergers only in the lower half of the MS. In our sample, major mergers on the MS occur with a roughly equal frequency of ̃5-10 per cent at all masses ≳ 1010 M☉. The MS major merger fraction roughly doubles between z = 0.2 and 2, with morphological mergers driving the overall increase at z ≳ 1. The differential redshift evolution of interacting pairs and morphologically classified mergers on the MS can be reconciled by evolving observability time-scales for both pairs and morphological disturbances. The observed variation of the late-stage merger fraction with ∆MS follows the perturbative 2-Star Formation Mode model, where any MS galaxy can experience a continuum of different star formation rate enhancements. This points to an SB-merger connection not only for extreme events, but also more moderate bursts which merely scatter galaxies upward within the MS, rather than fully elevating them above it.