How to get the shortest simple path in a directed Graph with an additional constraint that it needs to use two arcs in the said path
Let arc $a1$ be $(u_1, v_1)$ and $a2$ be $(u_2, v_2)$. As you mentioned in your question, you want to find a simple shortest path that: 


Goes from source $s$ to $u_1$
From $v_1$ to $u_2$
From $v_2$ to the sink $t$.


As you DAG has only positive weights, I think you should find the shortest paths for all three cases (using Dijkstra or any other algorithm) and join them together to get the shortest path from $s$ to $t$ satisfying the given constraints. 

EDIT: As the question is for Simple Path, I will refer you to this: http://math.mit.edu/~goemans/18438S12/lec22.pdf.

I am quoting a few important points here:


  
  We first note that the arc-disjoint path problem, and the vertex-disjoint path problem in a directed graph are equivalent - i.e. we can give a polynomial-time reduction in either direction. 
  Undirected Edge Disjoint Paths. In the case of fixed k (i.e. there are k source-sink pairs), there is a polynomial-time algorithm to decide if there are edge-disjoint paths connecting each $s_i$,$t_i$ pair. This result follows from the Graph Minor Project of Robertson and Seymour (1995), and is very involved.
  Yet for super-constant k, the undirected edge-disjoint paths problem is NP-complete. 22-1
  Directed Arc Disjoint Paths. Even for $k = 2$ it is NP-complete to determine if there are arc-disjoint paths P1, P2 connecting s1, t1 and s2, t2. In fact, we can even choose s1 = t2 and s2 = t1 and this problem still remains NP-complete.